We’ve not ridden ATVs in the area, but you’ll see signs nearby, and past house guests have ridden in the area. Brimstone on the Oneida side of the Big South Fork park is a popular area, but you’ll see You-Tubers posting about Picket State Forest which is very near Tanglewood.
Pickett State Forest abuts Picket State Park. The State Forest land permits off-road vehicles according to the road-use indicators on the map below