Haven't been there, but on our next trip East in 2017 we will plan a stop at the Virginia Highland Haven Airstream park in Copper Hill, Va. It gets good reviews. We stayed in Clemmons, NC (just west of Winston-Salem)in 2013 at Tanglewood RV Campground. Good facility and lots to do.
We have not driven the Parkway with a trailer but it would seem very feasible. We have stayed in the Asheville area many times (lived in NC for 40 years) and made day trips along the parkway. Rutledge Lake RV Resort in Fletcher, NC is a nice campground and is close to the Parkway, Biltmore, etc.
"If Mama ain't happy, ain't nobody happy" ... magnet on our refrigerator...