I remember the shorts you're talking about. TBS (and sometimes TNT) would air them when they needed to fill time after a movie ended early. Eventually, the shorts morphed into a regular half hour series called "The Chimp Channel", which had chimps running a network, interspersed with the parodies.

Here's what's weird about "The Chimp Channel": Comedy doesn't just happen magically; there are rules that make things funny, just like the fundamental laws of physics that make things happen in the natural world. The first rule of comedy is "Everything's funnier with a monkey". Corollary to rule 1: "There's nothing funnier than a monkey in people clothes".

Yet, even though these rules are unbreakable and fundamental to the order of the universe, "The Chimp Channel" succeeded in somehow being the most completely unfunny show of its time. I watched the first four episodes and didn't laugh at loud once. This was a severe disappointment to me as I thought the original shorts were pretty funny. And, I'm a huge fan of monkeys in people clothes.

Anyway, that's not really helping you to find the original shorts, but may give you more information for googling.