National Parks do not allow dogs on trails or anywhere other than paved areas & campgrounds. National Forests aren't as strict and state parks have the most lenient rules.
We do leave our dogs occasionally but not for long. At Zion, you have to take shuttles to see the park and of course, dogs can't go. We camped in the park at Watchman and fortunately had good shade & comfortable weather so left the furry ones for a couple of hours while we took the shuttles around. We made a trip back to check on them then caught the shuttle again.
I've not had any luck finding kennels that I would leave ours in.