I'm up closer to the north end of the Parkway, and I can tell you that entire sections of the Parkway close when there's snow on 'em. They do not plow, so if it snows, the gates get slammed shut.
For what it is worth, I believe the Lynchburg (VA) KOA at 6252 Elon Rd, 800-562-8717 is open all year. It is located 1 mile east of the Parkway on VA 130, and it is NOT "at the bottom of the mountain below the parkway." That stretch of the Parkway IS at the bottom of the hill.
This is about a half-hour south of Wintergreen Resort, a ski resort also about a mile off the Parkway, but the entrance is at the bottom of the mountain. Wintergreen also makes snow.