Technically firearms are not permitted in a state park.

If you take a chance on carrying it and someone in the park sees it and complains then you're in a tough spot as the park regulation specifically prohibits carrying a firearm.

However, waterfowl hunters go through the parks to get on their boats with firearms all the time. I would suggest you do the same thing as them. Don't carry the handgun on your person but have it in a locked case while transiting the park property.