Real names are only Google+ policy. In fact, I just checked, and a name isn't even required to have a Google Account. This is a bit of an edge case in that it is more of an account for a project than for a person, and Google doesn't really have it set up for this use (startup idea?). But it is the parent's account (they are entering the info, they write the emails from their perspective, input the pics/videos), so yes, the parent's birthday would be valid.

The real question is why did the OP even set up Google+ for an account intended to document their child's growth, and once they did, set it up as if the Google+ account was their child's.