The parking pawl is an aluminum finger that engages indentations on the drum. That is why when you try to put it in park while you are still moving it sounds like a stick being snapped along a picket fence. This looseness is more deeply internal like where the output shaft is inside somewhere. I am thinking someone did a lot of high rpm neutral-to-drive-dumps and ruined something. But I don't know enough about the T-350 innards.
I said BUT because a trans shouldn't do that in park period! Put your car in park and see how far it moves. The car up on jacks is the same as disconnecting the drive shaft.
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