I can't answer as a parent, but I can answer as a former theme park kid.
I was going to
Disneyland and Magic Mountain and walking around with just friends since I was 10 or 11. At Disneyland, which was an hour and a half from home, there would be a parent somewhere in the park, but not with us. Magic Mountain was 20 minutes from home, so we just got dropped off and picked up at the end of the day.
It was great fun to be able to wander without parents, and I never felt uneasy or uncomfortable. I felt very at home in both parks, so being with parents would have felt like being with parents at the mall or the roller rink — a lot less fun, and unnecessarily stifling.
As others have mentioned, all kids mature differently, but most 11-year-olds* would probably be fine wandering with friends (not solo), while by this time, the 16-year-old is probably way past antsy to be able to do it.
* I attended a camp during this period that took day trips to theme parks on occasion. The rule for 10- to 14-year-olds was that they could wander without a counselor as long as they were in groups of four or more. I mention this because while all kids are different, this is an age range that was widely enough understood to be okay having a bit of independence in a theme park that it was baked into the camp's rules. Granted, a group of four is not a group of two, but we also didn't have a 16-year-old with us.