If the child is not breaking any laws or harming other children, then no, the parent does not have any authority whatsoever on the information their child chooses to disclose to counselors and teachers.
The quickest way to get your kid to stop trusting you is to pry into the corners of their life that they don’t want you in. If a kid does not feel safe telling their parents about their sexual orientation or gender identity, there is 100% of the time a very good reason. And that reason is invariably some flavor of “I know my parents and this is NOT going to end well.”
As I said in another comment, the assumption that all parents are loving and want what’s best for their children is naive as hell. There are many, many parents — not just the Jesus Freak types, but plenty of narcissistic parents who want a Mini-Me rather than an independent child — who should not be permitted to care for a Pet Rock, nevermind another human being. And the best judge of how safe it is to tell a parent about LGBT issues is the kid who would be forced to live with that fallout.