My coach always said the worst part about her job was dealing with parents. We definitely had our fair share of hard-to-deal-with parents - parents who watch every 5-hour practice and stress their child out before meets, parents who don't make payment after payment, parents who are pushy with their child and with the coaches when it's clear to everyone else that the kid would rather not be doing gymnastics. But I think at the gym where I worked more recently, the relationships with parents were worse. The culture was different - parents (and some kids also) were more entitled and the gym was not very well managed, so communication was very difficult.
These were some of the issues that led to coaches not wanting to deal with parents:
*No time or method dedicated to parent/coach communication. I have 5 minutes between classes, I may or may not have time on any particular day to discuss your child's progress with you. If your child is in my last class of the day, you are lucky since I may be able to meet with you. We had no other way of communicating through the gym other than written status report sheets that unfortunately coaches also had no time to work on, especially in larger classes. I can't blame the parents for trying to reach out to me but it was a bad set-up.
*I had a relationship with the kid, not the parent(s). I had a good sense of each kid's personality, skills, and goals, so in most scenarios communicating directly with the student would be preferable. I think this is why you will hear coaches say things like "after a certain age, gymnastics is up to the gymnast, not the parent". I would communicate with parents about medical/behavioral problems, injuries/accidents, or if I thought the kid should be moved to a different class. Other than that I'd rather discuss things directly with the child if they're old enough. Of course sometimes kids are too shy to bring things up directly or don't know how to communicate it, in which case facilitation by the parent is helpful.
*Parents are sometimes out of step with the child's skills and goals. I've had a parent move a child who was NOT ready to the next class without consulting me (they can technically do this in our online class registration system, but most parents don't know that or don't try it). Another parent refused to move his child to a more age/developmentally-appropriate class since she was technically old enough for mine, but would have gotten more appropriate coaching and skill development in the younger class. Another time I had a kid who told me she didn't like gymnastics but her parents made her go, which I really don't understand the point of.
*Bad behavior parents: coaching from the sidelines, asserting (especially multiple times) that the child is ready to be moved up when they definitely aren't, letting siblings play on the equipment while they're waiting, parents playing on the equipment while they're waiting, parents who don't believe you when you discuss a behavior problem, parents who put children who aren't potty-trained into classes where that's required, parents who hit on you, parents who bring their badly-behaved dog to the gym, etc, etc. Of course not all parents are like this, but parents who initiate a lot of parent-coach interaction are more likely to be bad behavior parents.
*Sometimes, I didn't want to answer a parent's question just because I didn't like the answer I had to give. The gym I coached at had a pretty strict age limit for team. So when parents of an 8-year-old girl in Beginner 1 asked me about team, I had to explain our process for team selection, which begins at age 4 or 5 (maybe 6). I also didn't feel good about talking to parents whose kids would be moving to the team track, because I didn't like the way some of the main team coaches treated their gymnasts. That's why I don't work there anymore.