Continuity mistake: When the children are singing, "So Long, Farewell", as they are going upstairs they pass a picture of Lois' parents hanging along the middle of the stairwell. When they get to the top of the staircase, we see the exact same picture, but now at the head of the stairs.
Continuity mistake: In the part where Meg is asking Brian about God, and Brian answers he is an atheist, the camera pans out quickly, and we see an entrance to the kitchen with an air vent above it. Eventually the family discussion with Brian continues, and you see that the air vent is now gone.
Continuity mistake: When the crowd is outside Peter's house wanting Peter to heal them, a lady in pink comes out and says "It's him!". While she does this she is standing on the lawn, but in the next shot she is standing on the cement path. Other people in the crowd change position in this cut too.
Continuity mistake: Before Dr. Hartman delivers Bonnie's baby, he removes a small wheelchair and places it on a cabinet next to him. After delivering the baby, he makes a funny remark about Paula Poundstone, and then he hands the baby to Bonnie. Notice that the wheelchair and the cabinet are no longer present.