Can you camp with horses and trail ride in the park?

Hocking State Forest offers 40 miles of bridle trails and 23 horse campsites. You can find information on camping, bridle trail maps, and rules here: http://forestry.ohiodnr.gov/hocking#tabc1

There are camp grounds and trails all over the place though. I'm sure you can find places to ride.

Not at the state, no.