People do ride the parkway during the winter after its closed. This is different though. There is active road work to rebuild the road. The workers might not appreciate it. But then again... if you go on a weekend... they might not be there. Sorry to be vague, but ridding through is a personal call. I live on the other side of Asheville, so I don't know if people are riding it anyway. Calling the park officials will probably get you the official responce of its a closed road... maybe not. I've driven most that route but only ridden from Mitchel to asheville. For your perspective that will all be down hill. If you want to go to Mitchel, I would highly recomend it. Its one steep road however. Beyond that... Around Linville there are actually a few "flat" sections.
Have fun,
Scott