What are the symptoms? Do you hear one loud clunk from the starter but it doesn't spin the engine? If so, that's a sign of worn solenoid contacts in the starter. Did you put in a used one or a rebuilt one? This is a real common problem and the chances of a used starter doing the same thing are real likely.
If it does not make any loud clunk, listen for clicking of the starter relay while a helper turns the ignition switch. If it also does not click, bypass the relay with a jumper wire or by removing its cover, reinstalling it, and squeezing the contact. If it cranks when you squeeze it, suspect the engine computer. The neutral safety switch function is performed by the computer instead of a simple, reliable switch. Not sure how that helps the owner, but I ran into that with a '96 model still under warranty. Double check the shift lever. Be sure it's solidly in park or try it in neutral. Also be sure the indicator in the instrument cluster says "park". If it doesn't, suspect a problem with the "PRNDL" switch.
If the starter relay does click, watch the headlights or dome light. If they get real dim when trying to crank the engine, check the battery cables and particularly the cable connections to the battery posts.
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Monday, August 31st, 2020 AT 2:05 PM