I have heard plenty of men talk about their kids at work even being up during the night with them. The author is just an idiot who wants something to complain about. I haven’t seen women treated differently because of talking about their kids.
The discussion about children can easily spiral into a Mommy competition, showing off and bragging about each kid. That excludes non-parents and wears on a lot of guys.
Another risk women fail to appreciate is that always talking about their children at work subtly suggests that they aren’t mentally at work, or less capable professionally because of family responsibilities.