Westville or East Rock would be my top pics. East Rock has a lot of new parents but it’s a little transient, many are grad students who are only here a couple years. Westville has more long term residents, many young families, larger yards. I recommend renting in East Rock the first year, because it’s more densely populated and easier to make friends, and closer to downtown.

Wooster Square, Beaver Hills, Downtown, and Morris Cove are other family friendly neighborhoods in New Haven. Milford and Branford are nice nearby towns with small downtown areas.

My parents are also NW CT-ish and this is about an hour away, very easy.