If staying in Yellowstone there are no real "family" type parks as you might think of them. We stayed in Western Yellowstone (madison) for a week and the camping was rustic (clean bathrooms) but no electric or water. Water was available at the bathroom area and you just had to carry back to the camp ground. That area is bear country and they are very particular about water, food and such being left out. They had very nice evening ranger programs every night and had a great stream to swim in that ran right next to the campground. We were only about 35 minutes from old faithful. Showers were available at the Old Faithful Inn and in west yellowstone. We showered at the old faithful inn one night when some rain came in. Just head to the fron desk, tell them you need a shower and for $3 you get one. (or ask for a bath and they have some private rooms that have these large, claw foot bath tubs that looked great....In fact, they must have flet bad for us, or we looked that bad, that our kids 10 yo & 12yo showered for free. Although somewhat limiting, we found camping in the park great.
You didn't say what you're traveling with. Check out the campgrounds in the park, but don't wait too long to make reservations either in or outside the park. Fishing Bridge inside the park is centrally located, but you have to have a hard-sideed unit because of the bears. It sounds as if you're concerned about keeping the 15 yo busy. I remember those years when three daughters were all in their teens and they were looking for the social (boys) events available in big family parks. Don't think you'll find that in or near Yellowstone. However, there's something about Yellowstone that keeps even teenagers in awe. Have fun.