For opening dates of the National Park Service (NPS) campgrounds along the Blue Ridge Parkway, go to the Reserve America reservation site for each campground you are interested in. Scroll down the page. Toward the bottom you will see buttons for "season dates" and "booking window". Click on the "season dates" button and a popup window will show you the opening and closing dates.
This morning I checked on Julian Price near Blowing Rock, NC (Milepost 297) and Rocky Knob (Milepost 161) near Floyd, VA. Both campgrounds are open for walk-ins on May 1 and for season on May 8. Be aware most NPS campgrounds on the Blue Ridge Parkway do not have hookups. Many sites are small and tight given they were built years ago and more suited to pop-ups and tents than modern RV's. We live in NC and prefer camping at some of the beautiful private and state parks along the Parkway to the crowded and less well maintained NPS campgrounds.
As Sunnygirl indicated, Virginia Highland Haven Airstream Park near Copper Hill, VA. is located very close to the Blue Ridge Park Parkway (Milepost 136). This Airstream only campground is extremely well maintained with beautiful views of the Blue Ridge Mountains from the campsites. All of the 46 sites have full hookups and the campground is rarely fully booked. There is much to see and do in the surrounding area. If you are passing through Memorial Day weekend, there will be a hiking and wine tasting rally at the park open to all Airstreamers, sponsored by the Southeastern Camping Unit.
Below is a link to the Virginia Highland Haven website. The park is closed until May 1, but you can make reservations via email in the off season and there is no deposit required.
Virginia Highland Haven Airstream Park
In the Asheville NC area, a nice private campground near the Blue Ridge Parkway (Milepost 382.6) is Momma Gertie's Hideaway Campground in Swannanoa, NC. It is open year round and has some sites on a mountain with beautiful views. Here's a link:
Momma Gertie's Campground
The Blue Ridge Parkway skirts the edge of Stone Mountain State Park (Milepost 229) near Roaring Gap, NC. Stone Mountain has a campground with 41 electric and water sites. There is a dump station. The campground is open year-round.
Here is a link to the Stone Mountain State Park reservation page:
Stone Mountain State Park Camping Reservations