(Note that "to somebody's left (or right)" refers to how you are looking at them)

Rhonda's parents, Brooke (first seen in "Rhonda's Glasses") and Buckley (first seen in "Polishing Rhonda"), are in the upper-left corner, behind Rhonda
Nadine's parents are along the top, behind Nadine (later scenes show the woman standing next to Nadine; also, they appear with Nadine in "Stinky's Pumpkin")
Lila's father (first seen in "Ms. Perfect") is in the upper right corner, to Lila's right
Sid's parents are both to Sid's right (that's Sid's mother to the left of Lila - right after the ring toss, there's another scene of the families shown from the back where Sid and his parents are together; also, Sid's father is in "Fishing Trip")
I'm not sure who is between Sid and Stinky (behind Harold's mother, that is); I don't think it's Stinky's mother, as past episodes have strongly implied that Stinky lives with just his father; maybe Stinky has an "Aunt Stinky"?
Right below Rhonda is Brainy's mother, with Brainy's father to her right
Curly's mother and father are right next to Brainy's father (Curly is right behind Grandpa)
Harold's mother Marilyn (first seen in "Harold's Kitty") is to Harold's left, and his father Jerry is on his right (her dark hair was an animation goof, not a problem with the tape)
Sheena's mother is on Sheena's left (in front of Lila), and her father is on her right
Gerald's father Martin is standing behind Gerald (and next to Grandma)
Eugene's father is on Big Bob's left (he's seen in "Fishing Trip"; he's also seen from behind with a blonde woman who I assume is Eugene's mother, as she is seen where Eugene lives in "Eugene's Birthday")
Phoebe's father Kyo (first seen in "Phoebe Cheats") is in the lower right corner (that's Miriam Pataki standing next to him)

I don't know who the man with the blonde hair standing behind Lila is

Phoebe's mother (Reba), Stinky's father (first seen in "Helga's Boyfriend") and Gerald's mother (who still doesn't have a first name) appear in the episode, but don't appear in this picture