Regarding 4x4x4 PLL Parity:
Once you've gotten very used to recognizing all of the PLL cases, parity cases on 4x4x4 can simple by a matter or seeing a case that are much less or aren't familiar with. Because of this, you can almost instantly realize that you have PLL parity.
Somewhere between your recognition of PLLs and just remembering previous parity cases, you can also learn recognize what the PLL case would be without parity before you can perform any algorithms. It's very often that I can tell what post-parity PLL case I'll have before even performing the parity algorithm. In this way, it's like memorizing another set of PLL cases through recurrence in solves.
Aaand that's all I've got for this. Nice thread!
Edit: Actually, there's a similar "phenomenon" going on here regarding OLL parity. I can tell when a dedge is flipped before or during my last F2L pair. That's not quite a strange as the other feelings mentioned in the topic post, however.