We have traveled in a class c with 2 adults and 4 kids at Yellowstone and stayed at fishing bridge. In some ways its perfect. You have everything you need all the time. But as others have said parking can be a bugger. Depends on how crowded the park is. If school is still out for summer I would imagine there would be plenty of people.
But if I were you I would take your chances and use your Class C. You will have to park farther away, but I always have thought our Class C was perfect for touring things with the kids. Of course ours was only 27.5 feet. On that particular trip we were on the road for close to two weeks. We have traveled with kids in a truck pulling a 5th wheel and kids in a class C. I greatly preferred traveling with kids in the Class C. JMHO