In Episode 2747 (1990), Snuffy says that his mom and dad love to give grown-up Snuffleupagus parties all the time, and they're mostly about emptying and filling. Snuffy flashes back to scenes where his mother serves cabbage juice and cabbages, which are consumed by two off-screen snuffles.
In 1992, the producers of Sesame Street tested an episode called "Snuffy's Parents Get a Divorce". After a series of arguments, Daddy Snuffle moved out of the cave, and Snuffy and Alice had to deal with the emotional fallout of a breakup. The episode didn't test well with children, and the producers decided to shelve it. In the context of the show, Mommy and Daddy Snuffleupagus are still married. A photo from the episode can be seen in 40 Years of Life on the Street.
According to the book, Daddy Snuffle's puppet was the Snuffy puppet, with no changes or alterations. The two were never in a scene together.