I doubt he’s sold the copyright to the public. I don’t know why Bachmann would use any music without getting permission. Probably her locals did it without discussing it. Or maybe she got permission from the publisher and Petty just doesn’t know it.
There's no evidence her campaign didn't license the song from the appropriate legal entity. Petty would not have to know, just collect his regular check. He's scoring PR points because she's an eeeeeeevil Republican and he's a grand-standing jerk.
We see this dance every campaign cycle. Bob Dole couldn't use "Soul Man" even as a "Dole Man" parody. W couldn't use "Still the One." McCain had a problem with John Mellencamp over "Our Country."
A GOP campaign licenses song for public performance like any business, outraged "artist" threatens to sue, claims no permission given. Rinse, repeat.
Once you license your song to BMI or ASCAP, you get the benefit of the performance fees, but you lose the right to control who or under what circumstances someone else can use your song.