Any firearm, BB gun, air gun, CO2 gun, bow and arrow, or spear gun in possession in a state park must be unloaded or un-nocked and stored in a case or locked within a vehicle except when participating in an activity authorized by written permit.
Indiana's Turkey Run State Park - Rules
That is not the DNR website, it is a .com someone setup for the park. Look at the .gov site. It has similar wording but with the exclusion of LTCH holders. That verbiage sounds like the previous wording before the law passed to allow LTCH holders in the state parks.
Don't hold me to this but I think parts of turkey run are still closed due to flood damage, anything on the north side of the suspention bridge. I would recondmend shades its just down the road from turkey run.