It would be far easier and more fruitful if you got a wiring diagram and a voltmeter and when the starter will not crank, test for voltage at the starter solenoid. If there is no voltage, start working upstream, following the diagram, until you find voltage. You may also have to do voltage drop tests to see if the wiring can carry the current needed. Any corrosion on the battery pos and neg cables? Do some diagnosis and not just throw parts at it as what you have put on may be lower quality than what you took off.

 

I would go into the under dash fuse box and locate the 10a starter fuse. It might be 15a. Been so long since I went looking for that fuse.

Your ground cable has an aftermarket battery terminal at the battery. Did you replace the positive terminal also? If you did, the light blue wire that’s attached to the positive terminal goes to the starter.

You can also try to put the shifter in neutral by using the bypass switch on the shifter. If it starts in neutral, the computer is seeing the transmission as not being in park.

 

I did try to put the car in neutral, but that didn’t work. I haven’t replaced the positive battery terminal yet, as it looked fine to me. Regarding the fuses, I checked all of them, including the two under the dash. The smaller one was fine, but the starter relay fuse was melted (image attached). Even though I replaced it, the car would start normally a few times, but the problem would return.

I’m trying to figure out why the shaking causes the car to work every time. I tried moving the starter relay with my hand and cranking the car, but that didn’t help. I also tried checking the harness that connects to the engine, but that didn’t work either.