My kid wouldn’t be going somewhere I wasn’t allowed.
I think you underestimate how difficult some parents can be. There are a few notable skaters who were kicked out of my arena because of their parent's behaviour. It's easier to have a blanket 'no parents' rule than only allowing the well-behaved ones in. What they eventually did at my rink was ban them from the ice area, and isolated them to a specific viewing area. In another way, some kids get really distracted by their parents watching. I never allowed mine to come to training or competition because they were a distraction.
I'm glad having parents around works for Eteri's group though; the Russian training system is so tough, so it's good for them to have a mum or dad around.