My favorite example is Alan Sherman's "The Ballad of Harry Lewis," a parody of "The Battle Hymn of the Republic," whose chorus begins, "Glory, Glory Harry Lewis," and which includes these lines about the tailor who worked for Irving Roth --
O Harry Lewis perished
In the service of his Lord
He was trampling through the warehouse
Where the drapes of Roth are stored."
To be clear, this is perhaps mostly a personal view and (more importantly) isn't a value judgement. A comic song to existing music can be far more creative and entertaining than a mediocre parody. But I just view the category of the work differently.
I think Randy Rainbow's song are mostly funny lyrics, but also do generally overlap as song parodies, using the words of the originals to twist for comedic effect.
This version of Paul Simon's "The Sounds of Silence" is not only a parody song, but a wonderful parody. It's from a couple years ago by Don Caron and Linda Gower. There's a more recent sequel but I didn't find it nearly as good. It's funny, but mostly just funny words, not so much a parody. This, though is -- in its terrific writing...and performance.