Campgrounds near BRP in Va
BRP is a magical place. We've heard only good things about Virginia Highlands AS Park, although we haven't stayed there. We can also vouch for Fairystone State Park where we camped nearly a week in Summer 2015. We were very impressed with the Virginia State Park facilities in general --- they were clean, well situated, well staffed, and in good repair. We were disappointed with Peaks of Otter, a National Park, despite its location directly on the Parkway. Sites were readily available, as another reply points out, but facilities and campsites were in embarrassingly poor repair at the time of our visit and we found it a real challenge to find a site where we had a hope of getting level with our 25'. No hookups of any sort there.
There's also a very nice looking set of Army Corps of Engineers Campgrounds on Philpott Lake that is worth checking out. We drove in, but didn't camp there. Cheers.