They tend to be small startups that haven't had to "grow up" yet.
If you want to make it easier for my wife to pursue a career, you have to make it easier for me to take care of our daughter (without me quitting and becoming a stay-at-home dad). The sort of thing I am talking about is allowing a shorter work week, giving more vacation time, giving family leave that doesn't involve using my sick leave allowance, and so on. Above all, people need to understand that children (usually) have two parents. These things need to become normal.
We all work remotely, and sometimes at rather odd hours, to make this balance possible. It's not always easy, of course, but we figure it out.
If you're trying to actually have a presence in your kids' lives but no one else at work seems to have that facet in their lives, you're on your own, and your work (and peace of mind!) are going to suffer as you try to pass for a non-parent. But if you're working for people who know what you're dealing with, they can offer flexibility that non-parenting employees wouldn't need, share strategies and options, and trade stories.
This week, I've interviewed two candidates for senior developer roles who both have small children at home; I'm actually not sure if they were specifically interviewing with us because we're getting a rep for being parent-supportive, but that'd be interesting.
I can remember the other side of this; long before I ever thought about having kids, I remember when a fellow developer had his first child. He only took off for a week or so (we had 10 days vacation a year, and he stuck to that!), but then he would stumble into the office looking pale and his work went to hell. And of course his wife was home with the baby by herself all day, while he was at the office being mostly useless.
It didn't occur to me at the time that there might be a better way to handle it; though of course at the time it wasn't up to me anyway.
You're probably right that they are coming to you because you may have a reputation of being parent friendly.