Looking at a diagram it looks like the park pawl and the part it engages should be visible through the bottom.
One thing to look at is the linkage from the shifter to the parking pawl. There is a spring loaded piece on the end of the rod so that if the trans is turning it will ratchet (at least a little) Not good to do that though. If that spring is broken or weak it may not force that pawl to move enough to lock the output shaft into place.
If the pawl is not in place it won't engage either.
You should be able to see what is happening insideat the back of the transmission.
http://www.bobfist.com/Th350exploded.gif
