@VanPastorMan, I'm sure there might be some. But that's not the vast majority of what's going on here.
Some mental illness is genetic, some environmental, most a combo of both. Blaming a parent for a child's mental illness, though, is unfair and incorrect. They most likely didn't "cause" it in the way you are suggesting.
These parents are reacting to a real phenomena in their children. The cause of gender dysphoria can be varied. And the child might just be searching for a band aid on a much different problem (the most benign might be just plain old attention seeking. But it is also highly correlated with kids on the autism spectrum, and seen in kids with schizophrenia). But when it reaches the point of the child threatening or attempting to harm themselves because of the illness, the parents are put into a really tough spot. And if a medical or psychological expert tells you the best way to keep your kid alive is to allow them to have a surgery or give them these drugs, what are you supposed to do?
I know a lot of men who have mentally ill children--one had gender dysphoria diagnosed (in San Fran) and the dad pushed back. It led to a divorce (actually it sped it up--it was going to happen anyway). Daughter is still mentally ill, but outgrew the gender dysphoria. She attempted suicide a few times, though, and it was a living hell for his family. He was not a crazy parent and did not want his daughter to be a son.