I also suspect the starter as it has to be removed in order to remove the transmission. It may have been handled roughly and loosened a connection or perhaps it was the last straw that did it in. It's a real pain to do though so give yourself lots of time before starting this job.

Park neutral safety switch could be it but I don't remember even touching it when I swapped out my transmission. I did have to mess with the linkage going from the shifter to the transmission as after the first few drives the shifter would slip into neutral if I bumped it. It also slipped from "D" to "2" on its own on one occasion. I had lined up the linkage as before but it wasn't the same with the new transmission. This is an auto we are talking about correct? See if the next time you drive it and shift into park and turn the engine off, if you can move the shifter forwards more and hear it 'click' into place. If yes the linkage needs to be adjusted to the safety switch is engaged. Or manually push the shifter towards P while cranking and see if that does it.