As noted by Naturist, your issue may well be having sections of the Parkway closed due to snow. Generally, we don't get a lot of snow in our part of Virginia but if so, it will be at the higher elevations. IMO, the most scenic part of the BRP is in western NC and there are generally RV parks around Asheville and Boone open year round. In Virginia, there are several RV parks near the BRP in the Natural Bridge and Lexington area. Check rvparkreviews.com to get the scoop on those. At the BRP's most southern end near Cherokee, NC, there should be RV parks in Cherokee, Gatlinburg and Pigeon Forge open year round. Those areas are pretty much tourist traps on steroids but plenty to do near both the BRP and the entrance to the Smoky Mountains.
Wherever you travel, have fun and travel safely.