neutral and park should be the exact same "gear" if you will. the only difference is that in park the parking pawl is engaged.
is it possible that he motor itself locked up, broken compnent. any load placed on motor from then on would make it stop. was motor making any strange noises. does it make strange noises when you do get it to fire up in neutral?
it would be hard to diagnose unless you get it to a shop of some sort. doesnt sound like a easy driveway fix to me. once motor starts it should not be looking for a Park neutral switch anymore so that wouldnt "shut the motor off"
do some basic checks, fuel pressure, any faults in car? does motor spin over nice and easy when starter turns it? my 1992 when I bought it had a rod knock, that would come and go. but on cold starts it would turn over slowly and barely idle then when warmed up a bit the noise would come and motor ran like ~~~. stallig and such at different times. #2 bearing was completly welded to the crank. but that was my motor.