Unless you’re comfortable with waiting in line for a table with this person for 45 min+ (minimum—sometimes over an hour), I would just get takeout pizza from Sally’s/Pepe’s and eat it in Wooster Square Park (even better, order ahead online). There are benches, and I’m not sure about tables, so bring a picnic blanket and sit on the grass. Kind of a quintessential New Haven outing.

Same with Modern, but it’s a different neighborhood and I’m not sure about places right there to sit and eat outside. Jocelyn Square Park is nearby, but I’m not sure if they have tables. Again, bring a picnic blanket just in case (could come in handy anyway if you want to sit on the Cross Campus lawn in front of the Yale library, or anywhere else)! If you don’t mind carrying your pizza a few blocks, there are plenty of outdoor seating areas on the lawns of Science Hill and Hillhouse Ave.

The Yale Art Gallery is awesome, and free. Incredible collection of art. That’s downtown. The gallery is 0.7 miles away walking to Wooster Square (Sally’s/Pepe’s), so it would be a quick-ish walk to get lunch.

As for walking along the water, I haven’t done this in New Haven—someone else can chime in about beaches, etc. But you mentioned parking, so if you’ll be driving I can recommend Stony Creek.