When Peter joins Rudy and her friends in the house he is wearing black patent-leather shoes. When it is his time on the "bucking horse" with Cliff, he is wearing tennis shoes.
During the first part of season one Rudy is said to be five years old. In one episode it is repeated many times as Clair speaks to Rudy that someone is not a baby at the age of five. Later, during the slumber party episode she is said to be 4.
When Theo goes across the street to ask Peter's mom if he can stay for the slumber party, we see him get up and walk out. Cliff calls Peter over to him and while talking to Peter you can still see Theo sitting on the end table, but in the next shot we see Theo come back in from outside.
Correction to recently submitted goof. Theo can be seen in the background while the twins are singing their song but he is supposed to be across the street. In the next scene Theo walks back in.
When Cliff has Peter bouncing on his knee, there is a little toy piano on the table in one shot. In the next shot, the piano is gone.