They do allow to go up to paye meadows on private vehicles but highly recommend against doing it. Lots of idiots think that only 4x4 is the requirement for these steep climbs when actually driving skill is that matters most. I've myself seen a prado & fortuner crashing horribly on two different occasions. Not only did it resulted in loss of their liives but endangered everyone around them too.
As far as I know, only fairy meadows jeep track can't be done on private vehicle since that track was prepared by local villagers union funding & they're the only one maintaining it. Not an ideal situation but understandable especially considering that there's no parking place up there so why would anyone want to take his car there anyway.