You're likely to have the Parkway to yourself in winter as it's closed most of the time to cars, making it perfect for snowshoeing and cross country skiing after a good snow fall.
We don't get a ton of snow here in Virginia, but when we do, the Blue Ridge Parkway can be a great place to get out and enjoy a day of snowshoeing or cross country skiing. It's also a great opportunity to get your pup out and let him work out some energy as the chances of running into anyone are much lower than normal.
If there's snow on the ground, then the Parkway will almost certainly be closed to traffic. Park at any of the entrances and enjoy a day of very peaceful snowshoeing or skiing with plenty of scenic overlooks. More information about the Blue Ridge Parkway and where to access it can be found here.