My daughter has a pretty good coach, and he's a great guy. I was worried that he would suck, since a majority of coaches don't have a clue and can't be bothered to learn. And when my daughter started, I didn't know if she would like the sport so I just let her go to practice the first season and see what happened. If he had been a bad coach I'd probably have just advised her to try a different sport.
On our team were two phenomenal guys (think 4:15/1:52 level in 10th grade), but they had developed via a private coach. Just to get off main topic for a second, you have to remember that just a few years ago school running was cancelled due to COVID. So during that time these guys ran for a club and got really good; obviously they trusted their private coach and wanted to follow his program (a lot of people in the city did this, so outside coaching became pretty common). Our school coach was tolerant of the situation and flexible, but still expected them to come to practice and do his workouts, which they did, although sometimes they would work out on their own with his permission. In the off season our coach isn't available for daily practice because he lives too far from the school and has young kids, so our kids do their own thing. Those guys went to their club, so at the time my daughter joined the school team there was already some precedent of mixing an off season private program into the in season school program. In our case no private program, just my daughter and I working together on the long term plan.
Returning to my daughter's story, while I didn't interfere at all with my daughter's training or nag the coach about anything, I did notice a lot of bad habits my daughter had which were not really addressed, such as starting waaaay too fast in races, and trying to keep up with people way above her level in practice in ways that were counterproductive (e.g. running shakeout run the day before a race with an average HR of 190). I didn't fault the coach for this since she was just starting out and JV level at that time, but he doesn't really cover that side of things very well for any of the runners. So this is where I stepped in as a parent. Rather than say anything to the coach or be annoying, I just worked with my daughter directly via conversation at home, giving her some strategic advice for races (when she was still not sub 6:00 for the mile: "let everyone run the first 200m in 35 and then take your time reeling them in over next lap when they're gassed", etc). I took meticulous splits for my daughter and taught her how to adjust her pacing to get the best results. This led to very rapid huge improvements for her. She was still running too fast on easy days but the mileage was low so I just ignored that but continued to teach her why you want to run easy on your easy days and told her if she put in a good block of training over the summer, she'd likely close the gap on the friends she liked to run with and would soon be able to do so without it being a tempo run every day.
So I made my role to be teaching my daughter how to run and how to train, and let her coach run the actual training in season. In the off season I trained her and taught her not to set artificial limits, but to learn to test her limits while not overdoing it. She probably averaged 35 a week that summer but had a few weeks close to 50 and came back in XC to be #2 on the team and top 20 in the state. We've slowly increased mileage each off season, and now she's doing quite a lot, but I taught her not to be afraid of it while also not being stupid and doing any more than necessary.
As time has passed my daughter has continued to improve dramatically.. I continue to discuss things like strategy with her, keep her log for her which allows me to monitor patterns in her training that would be way too time consuming for her coach to do for any athlete, let alone the whole team. If I think she's getting overcooked, I advise her to run extra easy on the easy days, etc. She advocates for herself with her coach and I never get directly involved with the training during the season. My daughter and I work together in the off season to return to the next season with a vastly improved base and ready to train. My daughter now knows so much about training that she could coach herself, so I'm now just a second opinion. And she understands exactly how her coach's workouts are structured so she always goes to practice and runs the workout the way her coach designed it. In short I made her a student of the sport.
I think this kind of partnership between a parent and coach can work fantastically. The parent can pay attention to all the details specific to their own child (splits, mood, sleep, nutrition, attitude, overall fatigue level, etc) while the coach focuses on training them and building the team. I'd never be able to do all the things our coach does to keep the team going in the right direction, but I love my role as a parent and I'm super bummed that my daughter will eventually graduate and my role will end.