Yellowstone is not a pet-friendly park because of both the wildlife and the unstable ground. In the thermal areas, you must stay on the trail, because off it you risk falling through a thin crust into a pot of boiling acid, not a good way to die. You also can't let your dogs encounter and possibly tangle sith the native critters, some of which will eat said dogs for lunch (no feeding the wildlife) and some of which you dogs will try to eat for lunch. So you can't let your dogs off leash anywhere. You also can't take the on any trails -- for both those reasons plus they will attract wildlife you don't really want to encounter, grizzlys and wolves and mountain lions.