All people need some degree of confidence before they are able to accept criticism. This means that parents need to supervise carefully and break things down into steps so that a child can experience mostly success. Then they are able to accept constructive criticism in the small areas where they need more guidance. We all need some degree of confidence too before we are able to spend time with people that know more than us without feeling threatened.
Parental confidence also is important. I think often when parents themselves are insecure, they either project their own fears onto their children and make them insecure as well, or they convey to the children that there will not be acceptance unless the child does well. When a parent is confident, they will also look for the best people available to teach their child. When they are not, they will go for mediocre because they are unsure of the child's abilities or are intimidated by the skill level of the teacher in question.
One thing I've also found with students that I've had - the parents who are more involved and see themselves as the teacher at home have kids who are much more successful than those who leave the responsibility to the kids before they understand how to go about practicing or breaking down a problem into manageable pieces. Even though there does need to be a handing off of responsibility along the way, I think the process is a gradual one. It involves discussing a plan and then checking in during the practice, or writing down on a chart what was practiced and what specific things the child was working on improving etc until that process is very instinctual and natural. A 10 year old usually has no idea of how to go about breaking the problem into manageable chunks and that is where an adults life experience comes in, regardless of their ability in a particular area.
When I am teaching kids we do a lot of work on figuring out:
1) Where am I experiencing trouble?
2) Which hand is causing the problem, or what skill am I specifically missing?
3) How could I isolate that skill before reintegrating it into what I am trying to do?
4) What are several different ways of practicing that skill to make things more interesting?
5) How can I reintegrate the problem spot into the rest of what I am working on?