All humor aside, parenting a committed skater is the most thankless and misunderstood factor in the equation. And you don't have to be the parent of a skater who is legitimately heading to nationals to feel the brunt of the disdain for skating parents that is systemic to the profession, and tolerated or even encouraged by a lot of rinks and coaches.
If your child considers skating one of his or her main activities, you are a skating parent whether or not the child takes private lessons. It is the child's relationship to the sport that defines your role. Never let anyone denigrate your, your child's, or your family's commitment just because you choose not to take private lessons. And the reason for that is no one's business but your own. It might be time, it might be money, it might be dumblebunny. It's nobody's business unless you choose to share it.
Your decision to go group classes or synchro or privates, to test ISI or USFS, and whether or not to compete should be informed from the outset by one thing-- what can you afford. "Afford" means not just money, but also time and emotional investment. You can walk into it with a decision matrix at your back, where you've weighed every factor you can think of, or you can slide into it, as most of us do, suddenly finding yourself with an serious figure skater. But either way, always be aware of what you're spending, how much time you're giving, and how your skater is dealing with the various pressures of the sport, physical, emotional, and social.
Your skater's reasons for skating are another personal decision, and this is where a lot of parents and coaches get it wrong. I was the parent of an extremely gifted skater who did not like to compete. We pushed her into it, and she did quite well, but finally told us "done" when she was just sixteen. Whatever it is that skaters get from winning competitions, she just didn't care enough about the Gold, removing a lot of the motivation from the onerous training. Ironically, after she quit competing, she increased her training schedule. Because now she was doing it for her own reasons.
The reason for skating can change, as I've discussed before (search "goals" in the tag cloud). A skater doesn't have to make the decision to be an Olympian when they're five. A coach that tells you he'll get your 5 year old to the Olympics, or even to her Senior test, is talking through his hat. You can go with the flow and still not miss out on anything.
Start taking privates, if you do at all, when it feels right for you. In a strong program, a skater can learn an axel and a couple of doubles in class with extra practice. Privates will get them there faster, and probably with stronger technique, but it's not the only route. Again, let your emotions, your finances, and your schedule guide you. When you decide on privates, hire the coach that your skater likes. Don't worry too much about the end game. As you learn the discipline from the parent's point of view, you'll start learning how to recognize the signs that you need a new coach (ooo--great post topic, watch this space).
One place where I would resist the child's desire or lead is quitting. A lot of kids hit a snag, or have a fight with a friend at the rink, or get stuck at a skill and decide they want to quit. This is one of those times when I would push that child up that hill and help them see the vista from the top. They may decide to quit after all, but kids, especially young teens, are easily discouraged and then get embarrassed by their own discouragement. Help them through this. Remember that especially at the basic skills level (through Freestyle 1), and with kids younger than 9 or 10, you can quit and come back without losing much momentum; ditto with advanced skaters. They can pick up more or less where they left off. The danger zone is young teens in FS 3-6, where the difficulty jumps exponentially just as the hormones kick in. Without a compelling alternative, keep them in skating until they settle back down.
It's all too easy to slide into stereotype, to become a Stage Mother to your skater. Step outside yourself and watch. If you observe behavior that would make you cringe if you observed it in someone else, you've fallen for the evil stereotype.
Be the skating parent that makes all the coaches wish they had your kid.