There were a number of occasions in the 60's when Hanna-Barbera poked fun at popular music, and this is one of them. The "Bear" in the song title is Paw Rugg, patriarch of the Hillbilly Bears. If that laugh at the end sounds familiar, that's because Rugg's distinctive mumble & laugh are done by Henry Corden, who would later inherit the iconic role of Fred Flintstone from Alan Reed after Reed passed away in the 70's. Hypnopump uploaded "Do The Bear" to YouTube:
The rationale behind this was that H-B wanted to show that just about anyone could be a rock star, even for the shortest amount of time. In the swinging 60's, that was certainly true.