Parent's perspective:
Modern communication, instead of the flying piles of paper. Updated website. Team (password/gated) website with the handbook, meet schedules etc.. We don't want a paper copy of the handbook in June. 90% of parents take a picture of the various flying papers on their phones anyway.... Because they don't want the flying paper. Please save the paper and put it online where we can find it when we need it.
Interim communication via emails.
My most overriding concern as a parent is that you care about my kid as a person-- the person she is today, the person she is as a gymnast, the person she will be some day when she grows up. The rest is important too, but if she (and we parents) think you have her best interests in mind and care about her and her goals, that is half the battle.
Don't resort to verbally comparing the kids in a negative way, or challenge them to beat or keep up with so and so in a belittling way. I've even seen elite coaches doing this in videos. I think kids succeed despite that, not because of it. Most team gymnastics kids are pretty competitive already. They don't need you to pit them against each other with negative verbal taunts to fuel their competitive fires. That doesn't really work, and that's not the kind of person I am trying to teach my daughter to be. Fair, positive little or big competitions--go for it. She loves that stuff.
Try to not play favorites. It's the single biggest complaint I hear from my daughter and her teammates. Not the hard work. Not the conditioning. But the "why pretend to pick the best person to demonstrate this new drill, because with 'ABC coach' it always Suzie 1 or Suzie 2, never ever anyone else." One of our younger coaches just plain shows blatent favoritism to one or two kids. Their teammates, even the littles, do notice. And they think it stinks.
I am giving you my money, and my daughter is giving you her her trust, please be "present" with she is there. She loves corrections. She hates being ignored. The occasional small and sincere encouraging words from a coach are like nuggets of gold to her little self.
She may look like a big muscled mini superhero out there in the gym at times. But please remember that she is just a kid. A child. I am putting her in your hands. You would know what a little silly pumpkin she is if you could hear the car conversations. I swear these superhero-looking girls morph into the little girls they really are with each step out of the gym toward the car. Please don't ever forget that. Their childhood is a treasure and we are trusting you with a big chunk of it, please view that as the honor it is. I was given the privilege of raising and helping to guide this little soul, and I am sharing a little bit of that with you. I treasure that privilege, hope you do too.