I'm surprised no one so far has mentioned The Kipper Family (and, latterly, Sid Kipper on his own). To my mind, they're the most consistently brilliant parodists I've come across. Often their songs parody entire genres rather than individual songs. Their melodies are parodical too -- just different enough from the originals to sound like legitimate folkloric variants.
Nobody could come up with such a body of work without being very conversant with English traditional folksongs, and I get the impression that they love and respect the real thing; they're not making fun of it, but rather of people who take it too seriously (I'll say "folk snobs" to avoid running foul of that other thread...).
Lots of people come up with a brilliant parody from time to time, but the Kippers came out with album after album of them (and Sid Kipper continues to do so).
Of course, somebody who isn't familiar with English traditional songs will miss a lot of the points, but I think anyone who knows folksong in general will find enough there to make it worthwhile.