We were most recently at Yellowstone in June 2016. Between Yellowstone and Grand Teton we were in the area for the entire month plus the first week of July. It definitely got more crowded as the month went on and was crazy in Yellowstone over 4th of July weekend.
Memorial Day weekend comes early in 2020, so you will be there the last week of May. It should be much less crowded then as many families still have kids in school. In past trips we usually see very few families with school age kids until the first week of June and than it's more and more as the month goes on. We also have a 24 ft Class C and always drive it to tour the parks.
Entering the park early is a good suggestion, but I would add that later in the day is very good as well. We have noticed that many people head to the campgrounds or other lodging around dinner time, freeing up parking for our larger vehicle. The end of May is near the Summer solstice so there is a lot of daylight to tour the park. This
website shows that sunset is about 9:00PM the last week of May in West Yellowstone. Sunrise is about 5:45AM if you want an early start to the day.
We plan to be there again in 2020.