The text I linked originally says "or any other flying machine or aerial apparatus,". To me, that means drones. IANAL though blah blah.
I've contacted the State Parks office twice asking for a list of designated airfields. Crickets. Zero response either time.
The Metroparks in Cleveland area (by that I mean the Cleveland Metroparks and Lorain County Metroparks) both explicitly don't allow drones. I haven't tried to fly in either much, but on the 3-4 occasions, I've had zero problems and simply make sure to ask a ranger if I see one. In fact, I had one ranger sit in his car and watch for a bit. I thought he was going to come over and put the kibosh on it. He eventually walked over and simply wanted to see what drone I had as his son was thinking about getting into the hobby. He asked that I stay away from the crowds and wildlife and, otherwise, have fun. Very cool as have been all my interactions with LEO in Ohio.
Except for some hard up prison guards who got their panties in a bunch about me flying ~1 mile from a prison. I explained that I wasn't breaking any laws and why. And that I understood their concern and had no intention of coming any closer to the prison. Still didn't cut it. They wanted me to show some ID and I told them no and went about my business. They hung around a bit and then slinked away.
They were shocked when I explained that I could, 100% autonomously, fly the drone several miles and land it right in the middle of the prison yard if I wanted to. So, given that, if I was up to no good, why would I be standing in the open, on a major road with a bank and a gas station across the street that certainly had security cameras. They failed to see the logic.