The Tennessee State Park system has a "strict" policy concerning off-trail land use, including both letterboxing and geocaching. If anyone wishes to place a letterbox in any state park, they must first follow the proper permit procedure otherwise the letterbox will be pulled.

How do we know this? An employee e-mailed Green Tortuga after pulling two letterboxes that were planted without the proper permits. After inquiring about what the 'proper permit procedure' is, he did not get a response, so you might try asking around at the state park you're interested in until you find someone that can help.

Original source of policy

Yes, but there is a permit system. The permit application is for caches, but the guidelines for letterboxes would be the same. No one may "hide" an item on Metro Nashville government park property without permission. The geocaching community in Nashville came to the parks department for permission and went through a proposal and approval process. I take the applications and keep the paperwork on file for safety and security. A volunteer in the geocache community routinely checks the cache sites for the parks department for location accuracy, park and people safety, and container approval and reports back to me. Each approved cache has a Metro permit
affixed to it so Park Police know it is safe.

Nashville Area Parks
Original source of policy
Permit Application