Where do schools get the authority to instruct children on gender issues and sexuality?

For the most part, they don't do that. And certainly not in younger grades.

Now - there are some (primarily conservative) people who believe that depicting people who are gay or trans in literature, or acknowledging the existence of gay or trans students in the classroom (or even in the broader world) is "instructing children on gender issues and sexuality." This is probably incorrect, and I suspect that when courts get asked to interpret the law, those conservatives are going to get quite a surprise. After all, if showing a gay couple is instructing kids on sexuality, then so is showing a straight couple. If talking about a trans person is "instruction on gender issues," then so is talking about a cis person.

In later grades, almost all high schools teach classes in social studies, civics, biology, and some form of sex education. It is impossible to get a complete (to a high school level) understanding of those topics without being aware that gay and trans people exist, that their rights and place in society have been the subject of considerable contention, and helping students assess and incorporate that into their knowledge of the world. We expect high schoolers to handle some pretty mature things: to be made aware of where babies come from, to know about the horrors of war and genocide, etc. They can and should be able to handle learning that gay and trans folks are out there.