Things to get a coach on your good side. Get them a coffee if it's early. Don't pester them more than any other parent...however. Don't tell your kid to come over there when they are in workout/class and they are acting up( this really irks me ). Don't be dumb enough to talk negatively about them on the sidelines.
Offer them a drink while the kids are on break. Ok, I'm kidding. Really, I am. I just have a far off dream of coaching in a tropical place with surfer music in the background while I sip on a Mojito while I'm in shorts and barefoot and in an Hawaiian shirt. At least joke with them when you see the boys are being hellions ( aka boys ). Hopefully, you understand my point here.
Ok, back to being serious. Too early, too much paperwork. You shouldn't have to bribe a coach to get them on their side. I generally will decline gifts but sometimes I'll just shut up about it. That's a whole different issue, really.
I try to keep an open dialogue with all the parents with whom I have gymnasts I coach. Whether this is the 4 year old munchkin or team kid or a cheer tumbler or rec girl in an advanced beginner. Especially, as I'm not a social extrovert by my nature. However, in the gym I have to be this way. It's called good business sense. I'm competing for your dollar even indirectly so you don't take Johnny to baseball or Mandy to soccer instead. You have a right to hold your coaches accountable. Unfortunately, coaches have ego trips besides a zillion other things in their head when they are in the gym. And that's just when it's gym-related and they've left Life at the door!
Don't go threatening coaches of pulling your kid. Then Coach will get all defensive and generally will just nod and say ok. Sometimes, they'll dance afterwards and tell ya to go now not tomorrow.
However, this doesn't sound like your case. I started in a gym that was a fluffed up Mommy and Me gym and progressed from there to a bigger Rec gym with a small team to Competition Gyms and so on.
In my case, I probably try to talk to the Parents too often. I regret they receive my long winded emails and bulletins. I am dismayed when I never know their parents because they drop them off and pick them up outside.
I do believe as a Coach we have to extend ourselves to our gymnasts. This will vary depending on what they are there for, but I knew as a competitor ( in other sports ) that bonded more with the coaches that showed a small interest in giving a damn about me. I didn't hound the coach for attention and I probably was a pain in the butt. However, I'm a guy and there is a different bond between a male coach and male athlete than female coach and female athlete or opposite gender coach and athlete. Especially, given age ranges.
As for Private Lessons... I'd inquire about it to the coach in question. I put so much into my private lessons, that I really like to know both the parent and gymnast a bit before hand. As in I'm cautious of problematic parents or gymnasts ( i.e. crazy kids or pushy parents ) I want to know the goals they want out of the lessons. I've seen kids in rec get a PL for their bday or a gift or just because they want to get a cartwheel or bridge kickover. Never really saw the point of doing it when they are very young. I mainly like to have both the gymnast and parent on board with me so they will trust me a lot more which makes the whole process smoother. And because I can be quite grueling at times with private lessons.
I'll just say if I get a gymnast to bond with me, I can abuse them more ( I don't physically abuse them...besides the occasional pit throw though I do enjoy theraband snapping ) ; hahah. It means I can be tougher on critique ( though I still adhere to the compliment sandwich at times or the straight up approach of what is going [ wrong ] at the time ). If they're willing to put up with me they might be more willing to put up with my conditioning or slightly obsessive compulsive perfectionist side.
Keeping an open dialogue of communication can nip a lot of problems in the bud. However, I'm management besides being a team coach or have been the senior facilitative coach on the floor. This can differ my opinion versus when I've just been another coach on team rather than say my role as the coach of a recreational class/level and thus that gymnast's guide and coach ( besides being responsible for them ).