Have a bad BTSI solenoid. Is there a way to disable this so I can shift out of park and get to a repair shop

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:
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The brake pedal is depressed.

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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

is it a linkage issue? if the shift lever is working and you can feel the detents, check your fluid make sure it is full and clean, if not you could have internal issues with the trans

Does the engine start ? Is this a automatic transmission vehicle ? Do the brake lights work when you press on the brake pedal ? It probably has something to do with the Automatic Transmission Shift Lock
The brake transaxle shift interlock (BTSI) system consists of the body control module (BCM), powertrain control module (PCM), BTSI solenoid, stop lamp switch, and PRNDL selector switch. The BTSI system prevents the driver from moving the shift lever out of PARK unless the brake pedal is depressed while in RUN or CRANK.
The BCM controls the output to the BTSI solenoid by grounding the solenoid which energizes it, preventing shifting out of PARK. When the BCM receives input from the PCM over the Class II link that the transaxle is in PARK and the brake pedal is depressed, it will stop grounding the BTSI solenoid or de-energize it which will allow shifting. After the transaxle is moved out of the PARK position, the BCM shall monitor the status of the PRNDL serial data message sent by the PCM and shall continue to de-energize the solenoid until the transaxle has been returned to the PARK position.
In the event that the serial data messages are never received (approximately 5 seconds) by the BCM from the PCM, the BCM shall de-energize the solenoid to allow shifting. The transaxle can be shifted from PARK by placing the ignition switch in the ACC position. This eliminates power to the solenoid.

The park neutral safety switch is bad ! Put the vehicle in neutral an see if it starts then .
BTSI solenoid ???? What do you think this is ? I do believe you have this mixed up , it's really called shift interlock solenoid > a safety system that will not let the vehicle shift from park unless the driver is pushing the brake pedal .

Well , first off being stuck in gear, there probably is a problem with
The park lock solenoid prevents shifting out of the PARK position when the ignition switch is in LOCK, ACC, or OFF. The Brake Transmission Shift Interlock (BTSI) is designed to prevent shifting out of PARK when the ignition switch is in the RUN, BULB TEST, or START position, without pressing the brake pedal. The BTSI solenoid and the park lock solenoid are energized when the ignition switch is in the RUN, BULB TEST, or START position. When the park lock solenoid and the Brake Transmission Shift Interlock (BTSI) are energized, the mechanical lock connected to this park lock solenoid is pulled away from the shifter. The stop lamp switch receives ignition voltage in the RUN, BULB TEST, or START position from the BTSI Fuse and is normally closed, providing voltage through circuit 1135 to the BTSI solenoid, energizing the solenoid and mechanically locking the transmission shift lever whenever it is in PARK. When the brake pedal is pressed, the stop lamp switch opens, removing voltage from circuit 1135 to the BTSI solenoid. The BTSI solenoid is de-energized and the mechanical lock connected to the BTSI solenoid pulls back from the shifter allowing shifting.

You would do well to buy a service repair manual subscription from ALLDATA OEM Auto Repair Information for Professionals you have no idea of the complexity of the electronics on your vehicle . You will need wiring diagrams to sort out your electrical issues , suggest you watch videos on youtube for voltage drop testing . if your dad doesn't know electrical testing .

SHIFTER LOCKED IN PLACE

1. Remove the instrument panel cover below the steering column, then remove the steering column shrouds to gain access to the brake transmission shift interlock (BTSI) solenoid. The solenoid is part of the interlock cable assembly.
2. Disconnect the wire connector from the rear of the BTSI solenoid.
3. Insert the ignition key and turn it to the ON position.
4. With the brakes applied, try moving the shifter out of the PARK position.

• If the lever now moves freely in and out of PARK, perform the Circuit Test below.
• If the shift lever still does not move from the PARK position, remove the BTSI solenoid from the shift lever assembly. If the shift lever now moves freely in and out of the park position, the BTSI solenoid is faulty and the interlock cable must be replaced. If the shift lever still does not move from the park position, the problem is in the shift lever assembly or the ignition interlock cassette

some are set up with safety with brake lites out. ck if brake lites work if not turn key on do not start and shift to neautral and start and have brake lites repaired

Can't get out Park, starts ok, tried the manuel thing pressing
the pink release button, nothing. Had the car towed but
they had to hoist it on a truck. Was there a recall for this?

SHIFTER LOCKED IN PLACE

1. Remove the instrument panel cover below the steering column, then remove the steering column shrouds to gain access to the brake transmission shift interlock (BTSI) solenoid. The solenoid is part of the interlock cable assembly.
2. Disconnect the wire connector from the rear of the BTSI solenoid.
3. Insert the ignition key and turn it to the ON position.
4. With the brakes applied, try moving the shifter out of the PARK position.

• If the lever now moves freely in and out of PARK, perform the Circuit Test below.
• If the shift lever still does not move from the PARK position, remove the BTSI solenoid from the shift lever assembly. If the shift lever now moves freely in and out of the park position, the BTSI solenoid is faulty and the interlock cable must be replaced. If the shift lever still does not move from the park position, the problem is in the shift lever assembly or the ignition interlock cassette

center console shifter? if so, likely could be a misadj shift cable, you can pull loose top cover per instruction maneul tells you how to get out of park till you can get it adj, or repaired