Skating is a difficult sport through all stages. Coaches - even young ones - have usually been around a long time as competitors themselves. They have gone through the system; now, they are learning new skills. Along with that, they are learning how to deal with students - and how to deal with you. Teach your skater to say thank you after every lesson. Remind yourself to do the same. Saying "Thank you" goes a long way to opening the doors of communication between you and your coach, no matter what the level.
Most of all, respect your own position as the one who pays the bills. Do not be a Bully Boss. Do not think that you know more than the coach because you probably don't, unless you are a coach or judge yourself. But do not be bullied, either. Coaches are professionals. That is why you pay them. That does not give them the right to disrespect you, to talk down to you, ignore or patronize you as the parent. Know when your greatest asset, your child, needs something else or, in a worst-case scenario, is truly at physical or mental risk. Know that, at times, it is best to part company for the sake of everyone involved. Know, too, that walking away is not always the answer. Sometimes it leaves more questions.
You won't always make the right decisions. Just like the fact that there is no One-Size-Fits-All coach, there is no Easy Button and no book of instructions to help you maneuver the pitfalls. There are resources. USFS has a parents' committee. PSA (Professional Skaters Association of which all coaches must be members) has a coach committee. They both can offer invaluable resources and information to assist you. They both have sources for grievance if something is truly, seriously wrong. Be a smart consumer. Be a good employer. Most of all, be a great parent because when skating is finished, there is nothing like having your child respect you and say thanks for providing the support and opportunity to live out some or all of their dreams on ice.