I've gone around themeparks alone at a pretty young age, I guess. But I'm considered mature and responsible for my age. I could go around IOA and US alone at 14 years. However I would sometimes split at Busch Gardens a couple years younger than that.
I have felt safe at IOA and US. I split from my parents quite a bit now, and will be visiting both parks with my cousin who is also 15 years old and we'll be going around alone quite a bit, but I don't mind going with my parents some. But I usually want to see the Ghostbusters Show a bit more often than my parents do (er...I should say my DAD...Mom doesn't mind, but sometimes keeps Dad company) and they don't do coasters. We also each have a cell phone and contact each other every once in a while and we don't bother with the meeting up anymore unless we want to do something together, but we used to do that when I was younger.
If someone is 14 and older, no problem, as long as they are responsible and mature.