I was reviewing some of the park rules. They don't want people to go swimming in the rivers in the park, but they can't legally bar you from doing so, and end by suggesting some safety rules.
When many of the Eastern parks were established, the government had to make broad and very liberal allowances for access and use. Thus, there's no admission fees in the Great Smoky Mountains (yay!), and horses are allowed on most of the trails there (boo!). So I assume that even though the Park Service doesn't want the public to swim in Mammoth Cave National Park, they can't actually stop you from doing so.