I’ve done a bunch of six-person birthday dinners with parents in Yorkville, and the difference between a good night and a headache is almost always where the table sits, not which restaurant. Book early and say 'older parents, hard of hearing,' even if it feels like overkill.
On the quieter end, Café d’Alsace (Second Ave, upper 80s) is my usual pick. It’s bistro-style, takes reservations, and has wall booths that fit six without turning into a bench situation. On a Thursday at 6:30 you’ll mostly dodge the after-work bar crowd. The menu has enough plain/safe choices: roasted chicken, steak frites, spaetzle, not much heat. Avoid the tables right up front by the bar. If they try to seat you there, ask to wait for a booth—it’s worth ten minutes.
My second rec is Sanducci’s on First. It’s red-sauce Italian, not fancy, but the back room is noticeably calmer than the front room. Six-tops are normal there, portions are generous, and nobody’s going to rush you out. Chicken parm, baked ziti, that kind of thing. The front room near the bar gets loud, so if you can’t get the back, maybe pass.
A few I’d rule out for this specific group: Sfoglia is lovely but tight and expensive; Uva is a little far downtown and too date-night at peak; Pio Pio and most of the big-name Italian family-style spots can get shouty.
The single best move is to go for a later first seating or an early second seating—5:45 or 8:30—but if your group is like mine, 6:30 is the sweet spot. Just make the reservation by phone and confirm the table location.