I heard this parody a few years ago maybe on HBO or showtime. I can
only find one web page:
http://www.evilninja.net/blog/archives.php?id=A2002091

Here's the part listed:

The sun spread across the Bedrock sky
Like a bronto-burger laid out on a table
I am not Joe Rockhead,
Nor do I pretend to be;
Merely a stone-quarry worker,
Willing to bowl a frame or two

I grow old
I grow old
Shall I wear my saber-toothed tiger suit
Rolled?

In the cave as the women come through
Speaking of the Great Gazoo
Singing

Yabba

Dabba

Doo

I don't know that the web page has the exact wording correct.
Example-I remembered "saber tooth tiger pants rolled".

What I would like is the comedian's name, the complete text, and where
to find a copy of it.

Thanks

Not an answer, but here are a couple other transcriptions of the lyrics...

http://groups-beta.google.com/group/sci.lang.japan/msg/c20f46a692c0ae89
http://www.holyzoo.com/humor/classic/X0042_Love_Song_of_Flinsto.html