Yes all camper trailers including Pop Up campers and other RV's are allowed to drive through and stay in Yellowstone National Park.
To camp in your camper trailer or RV etc you must go to an RV campground in Yellowstone National Park and camp there and pay a fee each night you camp there.
The RV site rent can range from $40.00 to $55.00 per night depending on the campground and your camper trailer or RV size.
I've seen a few people with there camper trailers just parked in open areas of Yellowstone and they were dry camping.
So it's possible to dry camp in Yellowstone and I haven't seen any rules that say you can't dry camp in Yellowstone National Park.
The worse that would happen is a park ranger comes by and tells you that you can't camp there so you can try it if you can find a dry camping spot.