I just think it is very important to keep the roles separate. One of DD's coaches has two girls on the team, and over the years, I have seen what a terrific job she does in making sure her daughters understand when she is wearing her coach hat as opposed to being their mom. Because the gym does event coaching, she has coached them both on her event all the way up.
Over the years, I've come to believe that for kids to succeed and be healthy in this sport, the parental role is as important as the coaching role, but it is quite distinct. As a parent, my role is to support and encourage my child, to help her/him keep some perspective on the role of gymnastics in her/his life, and to serve as a sounding board or reality check when it's helpful. It is also my role to keep an eye on things in the gym to be sure that my child is being treated appropriately and that the program itself is continuing to be broadly functional, training and retaining good kids to reach relatively high levels.
I'm an insatiably curious and analytical type, so I've learned a lot about the sport and can talk somewhat knowledgeably with the kids and their coaches about what they're doing. However, I make it quite clear that it is NOT my role to critique or make suggestions about gymnastics skills or how the coaches teach them. The coaches need to have that clear and unambiguous line of authority to my child, and the child needs to understand that they are the experts here and that Mom and Dad are not the places to look for approval or suggestions. The relationship between the coach and the kid has to be direct, and I have no place in the middle of it or even on the side. While I will give advice on how to handle a coach in a bad mood or negative feedback if one of my children wants it, I am very careful never to undermine the authority of the coach as an expert on teaching gymnastics skills, and I say explicitly that the coach is the expert here, not me. My role in my kids' gymnastics is just as important, but it is very different. I think that it would be very hard to perform that role as it should be performed if I were sometimes acting as a coach without really being a designated coach for my child, and if I were a designated coach, I'd want to be really explicit and clear about when I was a coach and when I was a parent.