I've heard a couple of origin stories and both of them may be true. Doesn't matter. Shel Silverstein wrote the material and Waylon Jennings, Bobby Bare, Jerry Reed, and Mel Tillis became the "Old Dogs" to record it. It's a bit odd that Shel doesn't put in an appearance (he died in 1999 and the album was 1998). It's worth looking at the website bearing his name shelsilverstein.com to get an idea of just how widely he roamed in the realm of kids stuff. It's odd that it ignores some of his best material. it should at least have a pointer. The wikipedia article has a pretty good overview and it's well worth chasing down some of the more obscure material.
My introduction to Shel came from Playboy and if you listen to DrDemento, you'll frequently hear about the Great Smoke Off - which to my surprise can be found on youtube.