As far as I can tell, all the songs for the Cecille sketches are originals (not parodies of any other). Some novelty songwriters, though, have done original work that copies a popular style without using an existing tune directly. It made sense for "I'm Gonna Get to You" to be a Motown song, for instance...and for "Brush Down the Dogies*" to be country and western.
*Yes, that's "dogies" with one G, a term for cattle in some cowboy songs--though why the animator got inspired to compare teeth to cows, I'll probably never know.