I've heard that there are sections of the Appalachian Trail open to horse use, but other areas where they are not permitted at all. It's really intended for hikers, and there are some parts that simply are not safe or suitable for horses at all.
You could probably find other trails/rides that are open to horses along the entire length AND of equal or greater distance than the AT.
"Parelli horsemanship is just like painting by the numbers. You need absolutely no skill. You just put this color here and this color there, and when you're done, you have ... a mess no one wants." mp