Visible crew/equipment: When Dewey yells for Lois because the bow was put on the present, if you look to the left of Lois, you will see a crewmember kneeling down, handing Lois the Red Dress, who is holding a bucket of water. He quickly gets out of the camera's view. [This is on the DVD of the first season. Someone said on the audio commentary that after catching it, they deliberately left it in to see if anyone would spot it. It's still a mistake, but that's why it's there.]
Continuity mistake: When Lois is yelling at the boys for playing with fire, the location of the green scissors on the table keeps changing between shots.
Continuity mistake: When the boys are wrapping their parents present, Dewey is holding a bow in some shots and not holding one in other shots.
Other mistake: Reese is playing a Game Boy without a game in the slot.
Answer: Actually, the band is called They Might Be Giants; the song is called "Boss Of Me."
Xofer