Generally the lanes are 10-feet wide with grass shoulders. Yes, there are inclines. Elevation on the Parkway varies between 600 feet (around milepost 65) and 6000 feet (around milepost 430).
Depending on the height of your rig, a couple of the tunnels may require you driving in the center of the road. For example, here are some heights for the Bunches Bald Tunnel
Height at the edge line: 10' 6"
Height at the center line: 18' 4"
Height 1' from edge line: 11.98'
Height 2' from edge line: 13.31' - Driver's side is on center line.
Height 3' from edge line: 14.48'
Can you do it? Yes.
I would find it more enjoyable to find base camps off the Parkway and explore the Parkway in my tow vehicle.
Tom