All state parks including the Tennessee State Parks do allow pets in the state park and campground areas.
But all pets must remain on a leash and are not allowed to run free during the time in the state parks.
You must also never leave the pet unattended or unsupervised and may also upon request provide Proof of rabies shots and a physical description of the pet.
So make sure the pet has had it's rabies shots prior to taking the pet to the state park and also some areas of the parks such as picnic areas, swimming areas and food service areas may have a restriction on pets coming into the area.