No the BRP does not run into Tenn. However. The Parkway ends in Cherokee at the South end. And Tenn is a easy trip from there.
Honestly. In Oct you should have no problems road or other wise riding the Parkway. We have camped on the BRP in the higher elevations all the way to the closing date of the Campgrounds. Got cold a time or two. But never even close to being snowed in.
You will want to plan your trip carefully. there is only ONE place on the BRP in NC to buy gas. I believe it is near the Bluffs Over look. The rest of the time you will have to exit the Parkway to fuel, and as mentioned. Many of the roads leading to, and from it, are not too good.