I think that is correct, many of the national and state parks that we have been to west and south, like Land Between Lakes, Sequoia NP, etc, had specific campgrounds with horse facilities like corrals, troughs and access to trails. There are a few in the Midwest, like Governor Dodge State park, but nowhere near as many.
Of course, one needs to have up to date coggins, and horses should be current on wormers and vaccines, such as west nile virus.
I've never horse camped, and no longer own a horse, but I always wanted to.