What is up with Jesse and kids?

Jesse is analogous to Walt in this regard.

Walt's journey starts with the self-realization that his life has turned out pathetic, when compared to his potential when he was younger. Walt understands where it all went wrong (though considers it not his fault).

Jesse's parents, from what we see of them, very much raised him in a way that they treated him based on the worst parts of his behavior, rather than nurturing the good parts in Jesse.
This is a classic tale of parents making their child out to be a rebel at the first sight of stubbornness, treating him like he is becoming a problem child; and then not seeing that they caused their child to rebel against them.

Jesse blames his parents for making him who he is now. Although Breaking Bad's narrative style often leaves room for interpretation (and I can't conclusively say anything), I do think that Jesse correctly attributes blame to his parents for his childhood (at the very least his puberty and young adolescent years).

Just like how Walt gets treated as the failure that he grew up to be, and not the genius that he really is; Jesse gets treated like the white trash he has grown up to be, instead of the well-meaning kid that he really is.

Jesse hates who he is now. Although he cannot escape who he is, and in the first seasons he doesn't ostensibly struggle with self-loathing, that doesn't mean that he likes being where he has ended up.

To Jesse, a bad childhood (regardless of who's to blame) can be the foundation of a failed life. Therefore, he considered a child's parents as the only safeguard to make sure the child grows up to be happy.

Jesse also lives in an environment where parental neglect is so very common (junkies have a notorious habit of putting their family behind their need for a fix).

edit It's also possible that Jesse's parents were never told that they were parenting Jesse badly. They look like a happy middle class family, to an outside observer. And they seem to think of themselves as great parents, and attribute Jesse's problem to Jesse alone ("look at your brother, he's so much better than you!")
This may be further justification for Jesse telling parents to clean up their act. From Jesse's point of view, if someone had told his parents to reconsider their parenting style, maybe he wouldn't have grown up to be who he is now.

Not only is Jesse incentivized to warn parents of giving their children a bad childhood, he is also surrounded by people who are already giving their children a bad childhood.
As Jesse starts hating himself more and more, his desire to prevent others from suffering the same fate only increases.

Out of universe, you can also argue that Jesse's innate moral code about the innocence of children is proof that he is a good man inside.
Contrast that to Walt, who practically forced Walt Jr to drink tequila shots at a party, and not regretting that Walt Jr gets sick almost immediately. This scene is used to show that Walt's heart has changed, that he has become a bad man. If he had done that to a 12 year old, he would have been irrefutably evil. But Walt Jr is almost old enough to do these things on his own, so Walt's evil remains "arguable".

Jesse had a difficult childhood. His parent's weren't criminals but that doesn't make them good parents. They were hard and unforgiving towards him which made him feel alienated and unloved and an outcast.

Due to his own difficult childhood, he has a stronger sense of empathy toward other children who are growing up in difficult circumstances. He naturally would want to be the opposite of what he perceives his own parents to be.