I'm 99.9 percent sure they're open; I believe only closed days are Christmas and New year's day.... call to be sure. Weather could also/does affect closings.
Not only are they open, but they have a special 3 mile hiking tour led by park rangers from 10am to 2pm on the Saturday after Thanksgiving. The park should be open 8am to 6pm.
Look up GRANDFATHER MOUNTAIN STATE PARK IN NORTH CAROLINA. There should be a website to answer all questions about Holidays!
This might be on their website where the hours are located.
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