Hi. Just in NC I have hung at Pilot Mountain SP, South Mountains SP (backcountry), Stone Mountain SP (backcountry), Hanging Rock SP, Morrow Mountain SP (backcountry), Lake James SP and Lake Norman SP. I've never had a problem with any of these places. I'd say you're good to go in the State Park system. You might also check out Rock Castle Gorge,Va. and the Uwharrie National Forest, and not least, the Linville Gorge area. Hope this helps.
"Civilization is the limitless multiplication of unnecessary necessities." - Mark Twain
“I go to nature to be soothed and healed, and to have my senses put in order.” - John Burroughs