Tunnels - no problem but move over and hug the yellow line especially on the shorter ones to add clearance. All posted clearances are measured at the white line and all have an arc.
Stop often. We pull in to almost every over look. Watch for signs though for the couple you don't want to pull into if towing a trailer.
Visit blueridgeparkway.org for a lot more information, especially the list of hiking trails.
If you have plenty of time and can boondock, plan one night at each camprground, maybe two days at some. You will not regret it.
Must dos are the hike up to Devil's Courthouse, a meal at Pisgah Inn, Linville Falls, Mineral Museum, breakfast as Marby Mill, sunrise from at least one over look.
Enjoy the trip. I've done all of the parkway multiple times, but not all of it in one trip though it is on my bucket list.
2015 Ram 3500 4x4 Crew Cab SRW 6.4 Hemi LB 3.73 (12.4 hand calc avg mpg after 92,000 miles with camper)
2004 Lance 815 (prev: 2004 FW 35'; 1994 TT 30'; Tents)