Hi and welcome to FixYa, I am Kelly I just answered another post and your question appeared after my posting. I see it is 2 days old. I am right now looking up the engine and the wiring diagram.
To get ignition you must have:
1. A full oil level. The engine has a low oil level sensor that will kill ignition if the oil level is low.
2. A a working clutch brake pedal switch
3. Someone sitting in the seat.
4. PTO clutch dis-engaged
5. Key in run or start.
Without all of the above you will not get ignition and the start solenoid will not cycle. (Start solenoid does not go through the oil level sensor.)
From what I can tell this is the 19.5 Hp Briggs and Stratton 42E707-2631-E1
I tried crossing the model number to a manual and the closest I could get was the manual with the same model number as yours except the last digit is a 1 vs a 3 Manual link below.
Mower parts are here:
Note the diagrams under the picture. Each of those thumbnail pictures will open a new section.
I think once you sort out the switches that you will get both the start solenoid and the ignition to work.
Since this mower has been sitting for sometime you should probably check out the PTO clutch gap. There are 3 spring loaded bolts / nuts and there are 3 slots in the PTO clutch. Static the gap in the clutch must be 0.020. Otherwise you will not only have starting problems but blade engagement problems. Adjust the 3 bolts/nuts until you get a 0.020 gap. That should make the switches in the PTO clutch work. Just look at the wiring for any damage around the PTO clutch also... A previously thrown belt can trash the clutch wiring.
After the PTO clutch your checkin Brake.clutch pedal switch and the seat pedal switch.
For troubleshooting only you can disconnect ONE wire from the oil level sensor to see if you get spark.
I recommend you get the starter to work normally first and then troubleshoot ignition. Because except for the low oil level sensor they are common.
Safety switches can for troubleshooing only be bypassed by running a jumper wire from the white wire of the key switch to the starter solenoid white wire connection. If the starter works by doing this and does NOT work without the jumper... you have safety switch problems.
Hopefully with this information you can revive this mower from hibernation.
Feel free to ask any questions anytime I will stick with you here until we figure it out.
Thanks for choosing FixYa,