Many times it is the unsupervised kids that are cutting in line, being obnoxious, running, etc.
Really, I guess it is all a maturity level, not an actual age. I have two teens, one I would let go off by herself, the other no way, unless I was willing to let him get kicked out of a park, but I wouldn't want him to affect other people's vacations, so we always just make it a family trip, the whole thing. If my 7 year old wants to do something different than the teens, then I take her, and hubbie takes the older kids elsewhere.
It is not the safety issue I am worried about, it is how my kids would affect other people.
Just my two cents, for what it is worth.