When they say the park is open to private vehicles, they mean the road going up Zion Canyon. You will be restricted as to where you can park because of vehicle length, but you should be able to find a spot at the Grotto Picnic area and at the loop at the end of the road (there are a few RV spots there). I'm not sure how they handle parking in the winter at some of the smaller bus stops up the canyon. As noted, it can be pretty nice there even in November (we sometimes mountain bike around Thanksgiving further north in Moab). The road up Zion Canyon isn't too steep but one does have to pay attention for cars once the shuttle busses quit running.
If you are coming from LA, I assume you will enter Zion through Hurricane and Springdale off of I-15. You don't have to worry about the tunnel if you are coming from that direction. If you come from or exit the park to the east along UT 9 (say on the way to Bryce Canyon NP), only then will you have to worry about the tunnel. If you have to make arrangements at the entrance station, those are right on the road and you have to stop and pay your entrance fee there anyway. If you are on your way out and have to use the tunnel, just stop at the visitors center (walking distance from the campground) and talk to them when you're ready to go.