Who played the weird bird thing on SNL?

My wife an I can't remember the name of the weird bird character on SNL (not Will Farrell's Bird Family sketch). The only sketch I can specifically remember involved the bird thing on one of those quarter-rides outside of a department store. There was a dad who wanted his kid to ride it but the bird thing was masturbating on the ride. Are we both going crazy or was this actually a skit? It was probably in the mid-90s sometime.