Mike Matheny - the manager of the St. Louis Cardinals - wrote a letter to parents while he was coaching his son's little league team that touches on this idea. The whole thing is worth reading, but this is primarily his thesis:
"I believe that the biggest role of the parent is to be a silent source of encouragement. I think if you ask most boys what they would want their parents to do during the game; they would say "NOTHING". Once again, this is ALL about the boys. I believe that a little league parent feels that they must participate with loud cheering and "Come on, let's go, you can do it", which just adds more pressure to the kids. I will be putting plenty of pressure on these boys to play the game the right way with class, and respect, and they will put too much pressure on themselves and each other already. You as parents need to be the silent, constant, source of support."
Link to full letter: www.mac-n-seitz.com/teams/mike-matheny-letter.html
It is quite literally my job as a coach to point out a kids mistakes. They handle it really well nearly universally. But the overly critical parents (you can see them coming a long way off) cause so much negative reactions it makes my job harder.
And quite simply, anyone who has coached anything knows that being critical at the time of the event is almost universally detrimental (and not just for kids). Your mind/body is just not ready for coaching at the end of an adrenal dump event.
You need distance to evaluate performance, and the emotional burden of parent/kid relationships makes that more true not less.
Would you apply the same to, say, mathematics? Or chemistry? Or programming? Do you think the parents should not give their kids feedback on these subjects, and instead leave it to the teacher (aka technical coach)?
Look at it this way: you're starting a new job. A very stressful one, where you have to learn a bunch of new things, long days, and not just learning one thing - you're learning dozens of subject every day. There are a bunch of others in your cohort who are also learning and you're all compared and graded against each other.
Then you come home at night and you vent to your wife about your day and how this one guy is an ass-kisser and this other colleague is full of himself, with an example of something he said. And then your wife, who maybe took a college course on one of the topics you used as an example, says "yeah honey that sucks. BTW that example you just used, you're wrong, it's actually xyz". What would that accomplish? Would you think "oh thank you, now I didn't learn 50 new things today, but 51! Great!"? No, you'd think she massively missed the point, and is massively missing the point about you, and she would be.
I think that's the problem right there. When I was in school, the day was not long nor was it stressful. I suppose I'm inclined to agree with you that if the above circumstances are true, a parent should not behave that way. However, if those circumstances are true, and my kid seems at least average or above, I'd as a parent do some hard thinking and consider finding another school for him.
One thing I learned in all of my education: You may learn a lot when you are overloaded, but you learn nothing well. Not just at their level, but at university.
I'm good at a bunch of subjects - likely much better than the teacher (math, physics, etc). If school is so stressful that my kid cannot learn these well, and there is no room for help from me, then it's a bad school.
That's insane. Of course it is!
Until their skill in a subject surpasses yours, you're the only technical coach.
Which for all but one or two subjects, it will.
Look I'm not saying parents can't ever teach their children anything. What I'm saying is that parents need to know their place and role, and they can do much more good by being emotionally supportive and in general creating an environment in which children can and want to learn, than by being yet another instructor who's trying to cram ever more things into the child's head (much of which will be different from what their school or team coach is teaching them anyway).
I don't allow parents to do the "tiger mode" criticism on the sidelines because it is not objective. The parents have emotional investment in winning.