My username might be jazzfan, but you're dealing with one major Captain Beefheart fanatic here. I mean I go way back way with the Captain, as in I even got to see the original Magic Band when I was just a teenager and not by mistake mind you. I went to see the Captain Beefheart open for the J. Geils Band - he was booed off the stage but that didn't stop him from sending Zoot Horn Rollo back out to do a bass solo for an encore. Yeah, you could say me taste for weird runs long and deep.
Anyway more to the point. I don't get the connection between John Prine and Captain Beefheart. My first exposure to Beefheart was through Zappa, ala "Willie the Pimp" (I believe I still have my high school painting of Willie stashed away somewhere) and things have never been the same since. But Prine -> Beefheart???
By the way, I get the Prine -> Parsons connection and all the others.
If you don't already have it I highly recommend Pork Chop Blue Around the Rind
by the group Fast 'n' Bulbous
, a 2005 release on the Cuneiform label. The Captain's music brought wonderfully to life by Philip Johnston (of Microscopic Septet fame), Gary Lucas and company. And by all means, see this group live if you ever get the chance.