Can I coach from the sidelines?

Most clubs do not allow this. I watched a figure skating coach ask a parent to leave the rink because of this once. Depending on the club, you may be able to get away with coming down out of the stands to the boards for a few words every now and then but most figure skating clubs won't allow someone to stand at the boards instructing their child. If your kid is going to be on the ice for a fair bit of time without coaching, try giving them a list of skating skills that they can keep in their pocket. When they can't think of what to do next, they can consult their list. My daughter couldn't read more than a few words when she first started being on the ice without a coach so we drew some pictures to go with the words.

I heard an interview with an experienced sports psychologist talking about parents' roles in their childrens' sports and she made the point that each person has their role and if the parent is coaching, then who is doing the parenting. Coaches sometimes have to be a bit tough with a skater if they need a push to achieve a particular objective. That's part of their job. If the coach is being hard on the kid and the parents are focused on skating skills development, the kid may end up feeling like they're under attack from all sides. Two adults against one kid isn't fair. It's a perspective worth thinking about.