There are city parks, county parks, state parks, and national parks. I'm not aware of any hunting allowed in any of them in any state. TN, as well as most other states have National Forest land, not to be confused with National Parks, where hunting is allowed. They also have areas owned by the state where hunting is allowed. Most hunting is on private property that folks either own, or allow others to hunt on with permission.
Different states use different terminology. The State parks you refer to in NY sound like what we call WMA's, Wildlife Management Areas. here in GA. Lots of public land here and in TN. Just different terminology.