In addition to the National Park Service site, this website has a lot of good information- especially this table about tunnels. Note the only problems for tall RV's (5th wheels and motorhomes) are 3 bridges down around the south end of the parkway. Read the description of the two columns - it's the right hand column you have to pay attention to - since the tunnels are "arched" and bi-directional, you can't rely on being able to get into the middle of the road for the max height.
Tunnel Clearances - Blue Ridge Parkway
North of those bridges you see big tour busses on the parkway all the time.
Many of the campgrounds are quite small and the sites are small. But there are many commercial campgrounds right of many of the parkway exits. You just have to do some planning.
Buddy Ray - Atlanta
Jayco 2016 Eagle HT 26.5RLS
Ford 2016 F150 Lariat, 3.5L V6 Ecoboost
Max Tow Pkg, 36gal tank
Reese Sidewinder and Reese Titan 16k hitch