For what it's worth, I would plan the majority of my time in the Gatlinburg/Pigeon Forge/National Park area and then venture out from there. Unless you plan to drive the entire length of the parkway, I would just access it on a day trip out around cherokee, nc (south end of the park). And go up a little. Maybe as far as Asheville if you want. Besides the Biltmore, I think you will find that there is a lot more stuff (and sometimes the same activity) as there is in the Asheville area. There is so much to do around the Gatlinburg area and within a short drive from there to do.
As far as pulling the TT, I wouldn't have an issue with it. 21' would be easy on it. Your main concern will be heighth. Figure out the heighth of your TT (tallest point - AC, etc) and then go to
http://www.nps.gov/blri/planyourvisit/tunnel-heights.htm and see what the heights are there. You will see a minimum (sides) and maximum (in the middle). You may need to run it down the middle of some and just wait for traffic to slow before you do it.
Also, be advised there is usually some sort of construction and closures on the parkway. There will be detours that take you off the barkway and then back on at some point.