I have to say I love 'em - in the right context. We've got a bloody good parodist in our midst here, with some of Micca's lyrics hitting the spot every time.
His "The Hash my Father Scored" (It's in the Mudcat Songbook) once beautifully disarmed what was threatening to become a rather bitter piece of sectarian sniping in the bar of the Europa in Belfast when I was based there, and for that reason alone the song was worthwhile, IMNSHO.
Last year I spent ages learning Keith Marsden's "The King of Rome", only to hear Mike Gibson's wonderful parody about next-door's orange cat and its near-fatal taste for pigeons, and now I can't sing it when Mike's anywhere near for fear of cracking up in hysterics. His parody doesn't in any way detract from the original song - all it is is a very funny song in its own right which has particular appeal for those who know the original (and I'm still trying to get the words from Mike - when I do I'll post 'em here).
And what about all the "traditional" parodies - the 17th and 18th century piss-takes published as broadsides? Are they also somehow demeaning to the original and not to be sung? Of course not. A parody should stand or fall on its own merits - if it's a good song and funny or thought-provoking, then perform it. If it's a crock of crap, then let it quietly die.
My son recently introduced me to Weird Al Yankovic, and I love the guy - some of his parodies also had me in hysterics.
But the past tense is important - as mentioned, parodies are a bit like sticky chocolate pudding; once in a while they're fun, but they don't bear too much repetition.