I like Weird Al and Philemon Arthur & the Dung. In fact, PA&tD remind me of Weird Al's early days.
Does Arthur/Dung make fun of songs?
Nope, but they are two guys making funny songs with simple instruments. Just like Weird Al's early days. Al played the accordion and Bermuda (his drummer) banged some various things. In a similar fashion, Philemon Arthur played an untuned guitar or the accordion while the Dung banged various things.
Well, actually, most of their songs are just random nonsense, but some are clever satire disguised as random nonsense which parodies various stuff, sometimes including music. They also claim to be inspired by "everything from Chuck Berry to folk music".