During start Automatic gear does not change

selinoid ? Solenoid Wouldn't think so , but you never know without testing . With the truck running you step on the brake which let's you shift from park into a gear . That part works right ?
Automatic Transmission Shift Lock Control Circuit Description
The automatic transmission shift lock control system prevents the automatic transmission from being shifted out of park position unless you depress the brake pedal with the ignition in run or start. When the ignition switch is in run or start, the current flows from the BU/LPS fuse located in the engine wiring harness junction block through CKT 839 (PNK) to the park/neutral position switch. The park/neutral position switch also feeds the backup lamps portion of the switch. With the transmission in park, the current flows to the automatic transmission shift lock control fuse located in the engine wiring harness junction block through CKT 275 (LT GRN). The current then flows to the stoplamp switch through CKT 584 (LT GRN/BLK). The current continues to the automatic transmission shift lock control solenoid through CKT 1135 (DK GRN/WHT). The ground path is provided through CKT 1850 (BLK) to G203. When you depress the brake pedal, the normally closed contacts in the stoplamp switch open and the current flow is interrupted. When the current flow interrupts the automatic transmission shift lock control solenoid de-energizes. The transmission may then shift to another gear. Once the transmission shifts to another gear (except park), the contacts in the park/neutral switch are open thus stopping the current flow to energize the solenoid when the brake pedal releases.

Do you know how to test automotive electrical systems ? Know what a wiring diagram is ? How to use a DVOM - digital volt ohmmeter ? If not your best bet mite be to have a qualified repair shop check it out.
Your vehicle has a safety system that won't allow it to be taken out of park if the driver isn't pushing on the brake pedal . Could be the brake switch is bad , do the brake lights work ? Or the lock solenoid that keeps gear selector from moving could be bad or wiring problem .

Automatic Transmission Shift Lock Control Description and Operation

The automatic transmission park lock control system is a safety device that prevents an inadvertent shift out of PARK when the engine is running. The driver must press the brake pedal before moving the park lever out of the PARK position. The system consists of the following components:
• The automatic transmission park lock solenoid (serviced as the automatic transmission shift lock actuator), located within the floor shift control assembly with vehicles equipped with floor shift.
• The body control module, which controls the voltage supply circuit of the park lock control solenoid.
• The engine control module.
The body control module controls the voltage to the park lock control solenoid through the park lock control solenoid controlled voltage circuit. The following conditions must be met before the body control module will supply voltage to the park lock solenoid:
• The ignition is in the ON position.
• The engine control module sends an input via GMLAN serial data to the body control module indicating the transmission is in the PARK position.
• The body control module determines the brake pedal is applied according the brake pedal position.
Since the park lock control solenoid is permanently grounded, the body control module supplies voltage to the automatic transmission park lock control solenoid, unlocking the park lever allowing the driver to move the park lever out of the PARK position as the solenoid de-energizes. When the brake pedal is not applied, the body control module turns the control voltage output of the park lock control solenoid OFF, energizing the park lock control solenoid. The energized solenoid mechanically locks the park lever in the PARK position.
During remote start operation, the body control module will energize the park lock control circuit, locking the park lever in the PARK position.
The rear hatch could be just as complicated .

Rear Hatch/Gate Description and Operation
The liftgate can be opened by operating the switch on the instrument panel. Battery voltage is supplied to the liftgate actuator through the body control module (BCM). The BCM will pulse the liftgate actuator when a release input is received through the key fob, or the instrument panel switch.

Something wrong with the shift lock system maybe. Safety system that won't allow the gear selector to move from park unless the driver is pushing the brake pedal . Do your brake light's work ? Does your caddy have floor or column shift ?
Automatic Transmission Shift Lock Control Description and Operation (Floor Shift)
The automatic transmission shift lock control system is a safety device that prevents an inadvertent shift out of PARK when the engine is running. The driver must press the brake pedal before moving the shift lever out of the PARK position. The system consists of the following components:
• The automatic transmission shift lock solenoid (serviced as the automatic transmission shift lock actuator), as well as the body control module (BCM) and the engine control module (ECM).The shift lock solenoid is located within the floor shift control assembly with vehicles equipped with floor shift.
• The BCM controls the voltage supply circuit of the shift lock control solenoid. The following conditions must be before the BCM will supply voltage to the shift lock control solenoid.
The BCM controls the voltage to the shift lock control solenoid through the shift lock control solenoid controlled voltage circuit. The following conditions must be met before the BCM will supply voltage to the shift lock solenoid:
• The ignition is in the ON position.
• The ECM sends an input via GMLAN serial data to the BCM indicating the transmission is in the PARK position.
• The BCM determines the brake pedal is applied according the brake pedal position.
Since the shift lock control solenoid is permanently grounded, the BCM supplies voltage to the automatic transmission shift lock control solenoid, mechanically unlocking the shift lever allowing the driver to move the shift lever out of the PARK position as the solenoid energizes. When the brake pedal is not applied, the BCM turns the control voltage output of the shift lock control solenoid OFF, de-energizing the shift lock control solenoid. The de-energized solenoid mechanically locks the shift lever in the PARK position.
During remote start operation, the BCM will de-energize the shift lock control circuit, locking the shift lever in the PARK position.

Is it shifter on the floor ?
Ignition Lock Cylinder Control Actuator
If the vehicle is equipped with a floor mounted console gear shifter, it has a ignition lock cylinder control actuator system in the steering column. The ignition lock cylinder control actuator's purpose is to prevent the ignition key from being turned to the OFF position when the transmission is in gear and the vehicle may still be moving. The column ignition lock system consists of a ignition lock cylinder control actuator , and a park position switch that is located in the A/T shift lock control switch. The ignition lock cylinder control actuator contains a pin that is spring loaded out to mechanically prevent the ignition key cylinder from being turned to the lock position when vehicle transmission is not in the Park position. If vehicle power is lost, and/or the transmission is not in the Park position the operator will not be able to turn the ignition key to the lock position and will not be able to remove the ignition key from the column.

GM had problems with these ! Has a TSB - technical service bulletin !
Automatic Transmission Shift Lock Control Description and Operation
The automatic transmission shift lock control system is a safety device that prevents an inadvertent shift out of PARK when the engine is running. The driver must press the brake pedal before moving the shift lever out of the PARK position. The system consists of the following components:
• The automatic transmission shift lock solenoid (serviced as the automatic transmission shift lock actuator), as well as the body control module (BCM) and the engine control module (ECM). The shift lock solenoid is located within the floor shift control assembly with vehicles equipped with floor shift.
• The BCM controls the voltage to the shift lock control solenoid through the shift lock control solenoid controlled voltage circuit. The following conditions must be met before the BCM will remove voltage from the shift lock solenoid:
- The ignition is in the ON position.
- The engine control module (ECM) sends an input via GMLAN serial data to the BCM indicating the transmission is in the PARK position.
- The BCM determines the brake pedal is applied according the brake pedal position.
Since the shift lock control solenoid is permanently grounded, the BCM supplies voltage to the automatic transmission shift lock control solenoid, mechanically locking the shift lever in the PARK position as the solenoid energizes. When the brake pedal is applied, the BCM turns the control voltage output of the shift lock control solenoid OFF, de-energizing the shift lock control solenoid. The de-energized solenoid releases the mechanical lock allowing the driver to move the shift lever out of the PARK position. When the transmission is out of the PARK position, the shift lock control solenoid remains de-energized.
During remote start operation, the BCM will energize the shift lock control circuit, locking the shift lever in the PARK position.
Replaced shifter on a cobalt , broken plastic part inside , wouldn't press on micro switch to disengage ! Found replacement shifter at auto slavage yard .$20.00 US

Automatic Transmission Shift Lock Control Description
The brake transmission shift interlock (BTSI) system prevents the automatic transmission from being shifted out of PARK unless the brake pedal is depressed. When the brake pedal is depressed, the BTSI solenoid releases a lock pawl. This enables the shift linkage to move and the shift lever to operate.
When service operations are required, the BTSI system can be overridden.
On vehicles equipped with a column-mounted shift lever, override the BTSI system by performing the following steps:

1. Turn the ignition switch to the OFF position.
2. Remove battery voltage from the solenoid.
3. Turn the ignition switch to the ON position.

For vehicles equipped with floor-mounted shift levers, an override lever is mounted on the right side of the assembly. The override lever can be accessed by removing the shift lever boot. Push the lever to the left in order to override the BTSI.
Which does your S-10 have column or floor shift ? No relay ,has a solenoid .
With Column Shift A/T
The brake transmission shift interlock (BTSI) system prevents the automatic transmission from being shifted out of the PARK position unless the following conditions are present:
?€¢
The brake pedal is depressed.

?€¢
The ignition switch is turned to the RUN or the START position.

When the ignition switch is in the RUN or the START position, the current flows from the B/U Lamp fuse located in the underhood bussed electrical center (UBEC) through CKT 839 (PNK) to the park/neutral position switch. The park/neutral position switch also feeds the backup lamps portion of the switch. With the transmission in the PARK position, the current flows to the BTSI fuse located in the underhood bussed electrical center (UBEC) through CKT 275 (LT GRN). The current then flows to the TCC/stoplamp switch through CKT 584 (LT GRN/BLK). The current continues to the BTSI solenoid through CKT 1135 (DK GRN/WHT). The ground path is provided through CKT 1850 (BLK) to G207. When the brake pedal is depressed, the normally closed contacts in the TCC/stoplamp switch open and the current flow is interrupted. When the current flow is interrupted, the BTSI solenoid de-energizes. The transmission may then shift to another gear. When the transmission shifts to another gear (except PARK), the contacts in the park/neutral position switch open. This condition stops the current flow to energize the solenoid when the brake pedal releases.

Whic

I am mechanic I really need to no more make ect. but it could be battery or the ignition switch or ignition lock n key but like I said need more information this is a guess that I told you

check the shifter mechanism for binding up and not moving freely into positions. Not starting in park but in neutral indicates something is out of alignment with the shifter. May just need cleaning and a bit of oil/grease to fix.

that is park lock failure, build into the shifter section,it has an
electric solenoid lock. and a bypass button.
in the glove box, see oper. guide, it tells you where the erg. release
button location, and varies by sub body type and options.
its hidden cover, or a slot, on most cars. the slot type is to insert
any key in slot and push. bang, it opens (brake set,etc)

Test/Service/Replace

The circuit on newer cars with automatic transmissions has a neutral safety, or neutral start switch installed between the ignition switch and the starter. The ignition switch cannot operate the starter motor unless the transmission is in park or neutral.
Late model vehicles with manual transmissions have a starter/clutch interlock switch that requires the clutch to be depressed to the floor.
Neutral Safety Switch
Normal operations of a neutral safety switch provides a quick check for the adjustment of the gear selector linkage.

• To do this, move the selector slowly until it clicks into the PARK position.
• Turn the ignition key to the START position.
• If the starter operates, the PARK position is correct.
• After checking the PARK position, move the lever slowly toward the NEUTRAL position until the lever drops at the end of the NEUTRAL stop in the selector gate.
• If the starter also operates at this point, the gearshift linkage is properly adjusted.
• This quick test also tests the adjustment of the neutral safety switch.
• If the engine does not start in either or both of these positions, the neutral safety switch or the gear selector linkage needs adjustment or repair.

CAUTION
Since you must work under the vehicle to adjust most shift linkages make sure you properly raise and support the vehicle before working under it. Also, wear safety glasses or goggles while working under the vehicle.
A voltmeter can be used to check the switch for voltage when the ignition key is turned to the START position with the shift lever in PARK or NEUTRAL. If there is no voltage, the switch should be adjusted or replaced.

is a mechanical linkage or electic indacator if it linkage just adjust it ot take it to transmision shop they can adjust it for you