Hi there,
While having children can be a second job in and of itself, let's talk about a few things.
1) I am pretty sure you were joking about listing it as an EC but in case you weren't, no - that doesn't quite count. So if you put it there, even if you mean it light heartedly, its going to look strange.
2) It's not a negative - there are plenty of students without kids who don't engage with the class. Don't mistake a lower percentage of students with children as a sense that they tend to reject parents. They don't. Plain and simple.
3) How you mention it will be important. I know this is going to frustrate you but I'll say it anyway: I can't tell you if it's a good idea or not without knowing which schools and which essays and your overarching story fully. You might be able to mention it for some programs and not others. In some it might make more sense to put it in the heart of the application and for another as a check mark on the online form. It truly depends.
4) As for pulling an experience from fatherhood to write about - be careful here. It had better be an AMAZING example to even warrant consideration. We have never had our clients who have children write an essay about a parenting experience. That should hopefully give you an idea of the skepticism I'm exhibiting on this side of the keyboard.
Let me know if you have any questions and good luck!
Bhavik