“One of our first dads meetups was at the Whitney Museum,” said Mr. Schneider, a father of two. “There were five of us with our babies, having lunch. It was like we were one of the exhibits. We had people coming up to us saying, ‘What is this?’ ”

The good news is that the culture has started to catch up. In April, Daniel Murphy, the Mets second baseman, ignited a fury of radio chatter after missing two games to be at home with his wife and newborn son. When the chief executive of MongoDB Inc., a software firm, announced that he would step down this year to spend more time with his family, he noted that, “As a male C.E.O., I have been asked what kind of car I drive and what type of music I like, but never how I balance the demands of being both a dad and a C.E.O.”

Companies like Facebook and Change.org are among a group that have begun to offer generous paternity leave policies, and as a 2014 survey of dads revealed, 89 percent said it would be an important criterion in looking for a new job.

And then there are the fathers who are wearing their dad badges each day, proudly.

“Find somebody you don’t know and thank them for coming here today,” Mr. Watts prompted the conference attendees, each of whom got a gift bag (a water bottle and beef jerky) underneath his seat. Good-natured heckling was allowed, but only because each of these men shared a kind of understanding.

“You know, anybody who cares about their kid can learn to change diapers and do laundry and cook — and all of those things that are part of being an at-home parent,” Mr. O’Dowd said. “But you can’t overcome social stigma on your own. So while we want guys to feel like they can come here and bond with each other, we also want to enlighten the world a little bit. To say, you know, dads can do this gig, too.”