Though not always the same experience, many National Parks have adjacent National Forest lands and as many of you with dog companions know, dogs are welcome in National Forest areas, on the trails, campgrounds, etc. In our neck of the woods some of the less populated and thus more pleasant places to camp are on National Forest lands, with a view of the nearby National Park wonder, i.e. camping, canoeing at Mt. Adams, with nice views of Mt. St. Helens; or National Forest Lands adjacent to Mt. Rainier, etc. In the past (without a dog now) finding doggy friendly camp sites over the years led me to many wonderful spots in National Forests or on DNR lands.
I know this is off topic of dogs in National Parks, but thought I'd throw it in!