"A certain faction"? More than one faction, I think. I have my own strong opinions about which factions are worse, and yes there ARE better and worse factions... but there sure as hell isn't just one or even just two.
But none of this is about the rights of parents of whatever stripe. It's about the rights of the children, who are their own human beings and deserve to fully grow into their independence. And who are capable of sorting through a whole hell of a lot more than adults give them credit for, especially if you help them out a bit.
Your kid seeing a drag show isn't going to turn them gay. Somebody blathering the "good news" at your kid isn't going to turn them into a Jesus freak. NOTHING is going to influence your kid like that unless they get a LOT of it CONSTANTLY, AND been sheltered to the point where they never hear it challenged. The vulnerable kids are the ones who've never been asked to choose between real alternatives, never heard one adult contradict something another adult has told them, never been exposed to anything but indoctrination, never been given any critical thinking tools, never been taught to recognize when somebody's trying to persuade them, never really been allowed to question anything important, and have no resistance.
And even then they often get over it. You can brainwash a kid if you try hard enough, but it's a full time job and it's not results are not guaranteed.
The thing people really worry about is not kids getting other people's weird shit "shoved down their throats". It's that being exposed to the idea that there is weird shit out there might get their kids thinking about whether their OWN indoctrination falls into that category.