Yep, there really is a P.E.R.N. though it's a lot closer to Earth than Anne and Todd McCaffrey's dragon stories.
It's the Performance Equine Rescue Network. They rescue retired race horses and horses destined for slaughter and other unwanted horses.
I found it while doing a google for info on the books.
I suppose people who adopt those horses could call themselves "Horseriders of P.E.R.N.".