For some reason, I thought only the minyanim after a bris don’t say tachnun, but any beforehand do. I could be wrong on that.
A lot also depends on the level of the parents. If they’re Reform, there’s a good chance the baby isn’t even Jewish, as Reform have diluted their definition of what makes someone Jewish so much that a lot of their members aren’t real Jews. Conservative haven’t gone that far yet, thankfully.
Just call a Rav and ask next time!