That should make a great trip. Definitely pick up a copy of the book I mentioned. It's well worth the cost, less than $20 in paperback.
A few things to consider about riding on the BRP. There is virtually no development along it aside from a few park-approved lodges and shops. Camping areas are spread out (another reason for the book or at least a park map). To resupply food, you'll need to get off the parkway and ride down to towns along the way. Since the parkway runs right along the Blueridge, that means very significant elevation drops to the towns, with steep climbs back up. You'll need headlights and tail-lights for your bike. The park service requires it because of the tunnels, and you'll be glad you have them. Carry plenty of water bottles because you won't be able to get water just anywhere. Traffic is generally light on weekdays (except during fall leaf season) but can be heavy at times on weekends. Weather is much cooler and wetter on the parkway than roads at lower elevations due to the altitude. Bring rain gear.