There are many national parks and many many more facilities operated by the NPS. The feasibility of staying in them can vary a great deal. Some national parks have no road access at all.
For parks with developed campgrounds, a class A with a toad is hardly any less feasible than a trailer of the same general size. You often need to be prepared to dry camp. The biggest challenge is whether there are sites (and access roads) that can accommodate the RV, and that varies a good deal with how big the RV is. A 45' class A will be a lot more restrictive than a 30' or 35' one.
Towards the smaller end of the scale, my gut feeling is that there are suitably large spaces at most of the parks. I haven't done anything at all like a proper survey. So far, I have not generally had trouble finding sufficiently large sites with my 32' motorhome in national and state parks. That does not in any way mean that there are many sites I would not fit or fit comfortably in, only that I haven't usually had trouble finding and getting one the other sites.