First, the national parks issue. The law that was passed allows carrying in national parks per state law for the state the park is located in. OK, that seems pretty simple in and of itself.
Second, State Parks are exempted from being prohibited places. That also seems pretty simple.
Third, and the confusing part. You have to watch out for land that may be Department of Wildlife land inside a state park, and you also have to watch out for Army Corp of Engineer Land no matter where it is. Even though Department of Wildlife land may be inside the boundaries of a state park, that particular land is paid for by Department of Wildlife and is technically not part of the state park so different rules apply, mainly rules about hunting. The head state park ranger told me about the Department of Wildlife issue when I asked him about this.
In the instance of Greenleaf State Park, Camp Gruber is right next door so you have to be really careful there, some of the hiking trails will lead you on to Gruber (done that by accident).