From Muppet Wiki

A Bird Family appeared as animated characters in Follow That Bird. When told by Miss Finch of the possibility of having his own bird family, Big Bird has a daydream in which an animated version of himself spends time with a loving family of birds who all look like him. This dream was contradicted by the reality of Mommy Dodo, Daddy Dodo, Donny Dodo, and Marie Dodo.

The cartoon sequence was supplied by Toronto-based studio The Animation House Limited, who also animated the opening scene of Big Bird blowing up a balloon of the Warner Bros. logo.