I work in a Missouri state park, and i found nothing in my superintendent's policy book about simply carrying firearms inside a park, so i have to assume it's based on local (county) law, which would almost always allow for both open carry and licensed concealed carry (for visitors only; as a non-law-enforcement employee of the state, i'm under much more specific and crappier rules). However, unlike on Corps properties, hunting is not allowed inside state parks, and the discharge of any firearm has to be cleared with the superintendent of the park and is usually limited to black powder for war reenactment-style special events. So basically, treat it like you're in town. Don't be a jerk, and don't shoot at anyone that isn't trying to kill you, and you're unlikely to have any trouble.
Also, know what kind of park you're in before you carry in. You'd be surprised how many people confuse us with every other type of public use area in existence.