Please do not listen to anyone who recommends straddling/crossing the double yellow lines. They are there for a number of very good reasons and should never be crossed. Oncoming traffic, especially motorcycles, can come out of the suns glare in a heartbeat and they have every right to expect their lane won't be encroached upon by other vehicles.
You didn't set out your heights but from your profile pic I doubt many of the tunnels would be an issue. The two lowest are south of Asheville and can easily be avoided. You will probably draw the attention of the locals given that your coach is 30+ years old but there are lots of campgrounds on or near the Parkway.
