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I am planning a photography trip to Yellowstone in May-June. This will be my first time there, and I've done a lot of research. So far, I've planned 3 weeks, which seems to be on the short side to hit the prime wildlife and scenic areas--even if the weather is perfect. I suggest you choose one park for your 12-day window; otherwise, you will be doing too much driving and too little enjoying. Take into account that you are visiting when the parks are packed with other people and heavy traffic.
I also considered an RV because I wanted the freedom and convenience of easy stopping and over nighting in all kinds of remote places. That was a myth: the well developed parks have strict restrictions on RVs. In addition, I rejected an RV because it was more expensive, slower, and more cumbersome than the alternative of a regular vehicle and staying in motels. The RV rental is high, they eat gas, gas prices are high, RVs are slow, the mountain roads are narrow, parking for RVs is difficult to find in turnouts and other crowded tourist places, and RV camping places in the parks are expensive and booked long in advance. Plus, you often see RVs at motels so the inhabitants can get a decent shower, etc. So I opted for a regular vehicle as being cheaper, quicker, with fewer hassles, and giving me more maneuverability and freedom to see wildlife and landscapes.
If you Google "photographing" and the name of the park, you will find detailed info on the top photo spots at at these parks with lots of tips, including driving directions, light angles, and sample photos.