A Parent's Responsibilities

The most important thing for a skating parent to remember is that their skater is a person first, and a skater second. And in all probability the "person" in the skater is a young one, still growing, still maturing. Sometimes skaters seem so grown up, so easily able to handle pressure situations, that we forget they are still kids -- DON'T!!! The same kids that we see looking so grown-up on the competition ice probably still cuddle their favorite stuffed animals when they go to bed at night. Let them be kids, and support them as they grow.

Some other things for parents to think about

Balance: make sure there is balance in your skater's life. Allow time for school and personal growth. Very few skaters make skating their life career. Don't put so much focus on your child's skating that you forget they'll have to function in a "normal world" when they grow up. School is important. Social development is important. Being a kid is important.
Help to set goals: help your child to set his/her goals in skating. Help him or her to be realistic about those goals, but also to make them challenging enough to make the sport fulfilling. Make sure the goals are your SKATER's goals, not yours. Help to achieve these goals by setting targets, plotting progress, etc. Be willing to reevaluate when necessary.
Learn about the sport: learn enough about skating to recognize the elements. know enough to know when something is done well, and when it is done less well. Recognize progress. Be interested, and listen when your skater talks about progress or problems.
Support your Pro: pay your bills on time, get your skater to the rink on time. When you can't be there, make sure to tell the Pro in advance. Let the Pro participate in goal-setting discussions if possible; or if not, at least ensure that the Pro understands your skater's goals. Listen to your Pro's advice and instructions, and help to ensure that your skater follows those instructions when practicing or doing off-ice activities. Once the Pro understands your skater's goals, let the Pro have the freedom to design a program to achieve them -- don't try to second-guess the approach. On the other hand, responsibly watch to make sure that the general goals are being addressed over the long term.
Watch: don't use the rink as a babysitter. Stay and watch your skater practice, and in lessons, at least sometimes. They need to know you're interested. Often skaters whose parents never watch in practice feel very self-conscious or "pressured" when their parents finally do show up to watch. If competition events are the only time you watch your skater, you may be hurting them more than helping them...
Be a good sport: Remember that every skater is someone's child, and that they ALL deserve to be treated fairly and with respect. Be willing to recognize other skaters graciously. When someone else is skating in a competition, don't walk back and forth in the bleachers, don't be noisy or disruptive.
Support your skater: Remember, your skater is still maturing. Offer praise when appropriate, but be realistic with that praise; recognize progress towards goals, but be willing to acknowledge when more work is needed without making that a "bad thing". Never destructively criticize, especially in front of others. Resist the urge to compare your child against another. Some learn faster, some learn slower.
Support your Club: pay your bills on time. Help with club activities. Most clubs are run by volunteers, and they need all the help they can get. Be willing to work on committees, or serve on the Board of Directors if needed. Be willing to do "jobs" like ice monitor, music monitor, etc. Help out with ice shows or competitions.