Definitely avoid pulling your TT on the Blue Ridge Parkway because at that time of year (and also in autumn) traffic is so heavy that its often crawling or at a stand still. The suggestion to use your tow vehicle to see sights on the Parkway is spot on for that reason. Find a campground you can return to at the end of the day and leave your trailer there as a base camp so that you can do spontaneous stops if the mood hits you to see a place that might not accommodate a trailer in tow.
Among the many sights and experiences to see and do:
-train ride and train museum in Bryson City, NC
-Fontana Dam in Fontana NC
-Joyce Kilmer Memorial Forest outside of Robbinsville, NC
Note: we prefer the western NC side of the Smoky Mtns as it's less noisy, less of a tourist-trap feel than Gatlinburg area and more emphasis on culture and history on the NC side. Bryson City and Asheville both have excellent restaurants and wine retailers.
Go to this link to see more activities and attractions on the NC side of the Smokys. Many of them are not crowded either as EVERYONE tends to flock to Cades Cove, Gatlinburg and Pigeon Forge on the TN side of the Smokys.
http://www.westernncattractions.com/