They have absolutely no control whatsoever over the interior. All they can do is say "park" or "don't park". That's it.
I think the courts would disagree once an employer started such nonsense.
That's probably true, but not very interesting; the court has a very sketchy history of upholding property rights--not to mention other civil rights.
...justifiably so.
In other words, you believe in taking away people's property rights unless you agree with their decisions? Nekulturny, Kamarad!
As an employer I stick with my post, and yes I believe the courts would laugh at any of my attempts to take control of my employees lives by taking control of the interiors of their automobiles. I believe that an employee of Wal Mart could win a case allowing her to keep her baby seat in the car while at work.