One State Park, Falls of the Ohio, is no-go due to it being ACoE-controlled.
State Parks with Reservoirs are another legal sticky wicket. Carry is legal at the State Parks owned/controlled portions of those properties, but if the Reservoir is ACoE controlled, then the lake itself + several feet of elevation from the water surface is also ACoE controlled - and therefore off limits on a Federal level.
A FEW State Properties are now State Historic Sites and have somehow managed to finagle a loophole and are allowed to have a RULE against handguns/firearms on their property. Mounds is now a Historic Site and falls into this new category. RULE - not a law.
Just like before the State Park RULE was changed. It was a DNR administrative rule - not a LAW. In "recent times", it has always been legal to carry at a State Park - provided you're properly credentialed.
Surly Rangers are also considered a no-no even if they provoke the action. However, they are rarely venomous and some of the female species are rather attractive if you stand back and admire them from a distance.