I agree with what's been said here. Helicopter parents drive me bonkers. When I was young, we called them "rink mothers". Those were the ones who were always around and in everyone's business, usually in a disruptive way.
I feel bad for the young lady on 4711's troop. I bet if left to her own devices, she'd have preferred to stay in GS and without her mother. It reminds me of a situation I was teaching high school The students with helicopter mothers were generally the least successful. I even know of one who after college was required to go home and live with her parents, and not allowed to drive, so really no chance for a future. She's not strong enough to walk away. It's very sad.
Speaking as a parent, I agree with parents being kept at a distance. If they must stay at the rink, then perhaps restricted to some area behind plexiglass, if not away from the ice altogether. However, I would make an exception for little kids. Mine are 3 and 5. The 3yo has a 1/2 hour skating lesson when she wants to go. But my 5yo plays hockey. I don't feel any personal need to observe, but do sit rinkside still, simply because they both need help with getting off the ice and going potty, and just in case there's an injury. Plus, my 3yo needs to actually see me or she freaks out. I am careful to NOT coach/comment/interfere from the boards. And I stay out of it when the hockey coach disciplines my son (the coaches don't discipline the 3/4yo's). This year, it would be the hockey coaches that would be bothered by my presence, and they don't seem to be. All the other parents are there, too though. I am SO looking forward to my kids being old enough for me to either stay in the warming room, or better yet, drop and go!