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This is the only time I'm going to bump this. I'm kind of losing hope on ever finding it
You could try looking on the REM song's wikipedia page to see if they list parodies
It's the end of the world as we know it and I feel fine?
Sanity is for the weak
^ A parody of that.
The local radio station could have made their own parody, so you could email them. If you don't know the radio station, you could just email all the radio stations and see if any of them remember. Might take a little time to see a reponse, though. You could also see if there are any local forums from that town and ask around. Or post in a forum for a big city in Michigan asking if anyone who used to live around Marquette (or knows someone who does) and if they remember the song you mean.
This is weird, because I feel like this must be regional at this point. It was a parody of "It's the end of the world as we know it (and I feel fine)" by REM that was something like "it's the end of the week and I know it" or something. I used to live up in Marquette, Michigan and a radio station somewhere around there would play it every Friday morning. Also among the words that I can remember are "it always starts on Friday". I've been looking for this for years, with little luck. I finally found proof that it exists, but now even that is gone.