These responses are interesting to me, I guess I was just lucky as a kid.
My mom dropped me off 30 minutes before my lesson (to tack up), came back about 15 minutes after the lesson finished. The time between the drop off and pick up was about learning more than riding - it was also about learning how to interact with people and gain independence.
No parents were there. Instructors and maybe a couple barn hands were around to help. If I had any problems, it was between the instructor and me, my mom was never there to step in. My instructor had all the tools she needed (amazing what cleaning extra tack, or not being able to jump that lesson, might do to make a kid behave).
Only if something went beyond my instructor's ability to handle would she have a "talk" with my mom after the lesson. (And you'd better believe that if she had a "talk" with any of our parents, we would be in a lot of trouble when we got home later.)
Every so often we'd have a parents' day where we'd get to show off - "teach" our parents about how to groom the horses, have them watch us ride, take tons of pictures. But not during the normal lessons.
I wouldn't trade the knowledge I gained from that experience for the world. It was a chance to step away from my mom and learn some independence.
In addition, even if the parents don't get in the way, the kids often still act different. Haven't you ever seen a teen who is polite as can be until mom walks up, and then all of a sudden turns snotty? That attitude makes it into the ring, too.
The last place I boarded (before I moved) was the same way as how I grew up. The instructor wouldn't let parents hang around except on special days. Any other time, kids were dropped off and picked up later. Amazing how well the kids behaved.
I guess I just thought all barns were like that...