According to http://www.buckowens.com/discography.html:

There was a 45-rpm single, Warner Bros. 8433:
OUR OLD MANSION/HOW COME MY DOG DON'T BARK, released August 3, 1977.

It was also on the album "Our Old Mansion" Warner Brothers BS3087, released Sept. 30, 1977.

It's also on "The Complete Warner Bros. Recordings" a 2-CD set (Rhino 2010).

According to repertoire.BMI.com, Buck Owens was the sole writer of HOW COME MY DOG DON'T BARK.

This agrees with the information at the U.S. Copyright Office.

I finally found a sound sample at Rhino Records:

"...Timbucktoo.
Uncle Bill's in California.
Cousin Will's sick with pneumonia.
Yeah, an' that don't leave no one but me and you.
How come my dog don't bark when you come 'round?
He wags his tail and lays down on the ground...."

Therefore I think we can safely say it's not the same song as any of those in the other thread.