Star Trekking by The Firm I believe.
I love quite a few of the songs already posted. Kip Adotta's Wet Dream is a genius song. I love how a lot of people discredit novelty/funny songs but don't realize how much wordplay comes into writing them. Obviously Weird Al opened many doors in the 80s but the doors were already open. Stan Freberg had some classics, even if some don't consider them songs more than talking. I'd be curious when everyone's "prime years" of listening was. For me it was the late 90s, so it was a mix of some of the all time best, but also a new age of novelty songs. By then Weird Al was still chugging along but after hearing Dr. Demento sometime around 95 or so, I was much more into the likes of Tim Cavanaugh (99 Dead Baboons, Really Safe Sex), The Frantics (Last Will & Temperament, Boot To The Head), The Arrogant Worms (Me Like Hockey, Big Fat Road Manager), as well as so many others. I think it was around 2001 or so where it was getting tougher to find the show, and the new stuff wasn't as appealing so I stopped listening. Travis Meyer's The Devil Went Down To Jamaica was one of the last hits I remember. It was also around that time I started listening to Christian music more and there was a band labeled as "That Christian Parody Band" who would rework famous songs into biblical stories and Bible verses. The ban is named Apologetix, and while I don't listen to them or much Christian music today, they still make albums. I don't know if they ever made it on Dr. Demento, but it would have fit.
For those who don't have any inkling to who Dr. Demnto is, the Weird Al movie does feature him played by The Office's Rain Wilson. There is a website that features all the old playlists from Dr. Demento's shows, as well- http://dmdb.org/playlists/ My favorite feature on the show was the news with Whimsical Will who also did songs on the blockbuster movie of the summer, in a style I tried to replicate a few times on projects, but never as good as him.
DON'T FORGET TO STAYYYYYYYYY DEMENTED!
Royal Card Review is my blog if you feel like checking it out, thanks if you do!- royalcardreview.blogspot.com/
In the process of updating my collection so don't trust any of my lists right now.