Won't crank in Park but will in nuetral

You might want to look into the seatbelt interlock, and take it out or bypass it if your car has one.
You can find this info here with a search.
They can do this due to a bad switch on the trans also.

I NEVER EVER leave my car in gear, when I park and I always take a glance to make sure it's in park before I hit the key, so I wouldn't hesitate to remove the safety switch from the starting system completely on my own car if I had a problem with it.

Besides, it's nice if the car stalls for some reason then you can just hit the key again without having to get it in neutral rolling, or stop to use park to restart.

The specific problem you are having does sound like the switch on the trans though, (IF you have a column shifted trans)
Consol shifted sometimes uses a switch on the shifter mechanism, and they can also work in one position and not the other due to adjustment issue's.

There is no adjustment that will help if the switch is in the side of the trans, as it relies on the linkage inside the trans to determine whether it is in park, neutral or other.
It's pretty much always a bad switch on those if it works for one or the other but not both park and neutral.(unless someone mixed and matched parts inside the trans)