Getting legal clearance for samples and covers can be a real problem. For samples, if the copyright owner of the sampled song discovers you've used it without permission, they can sue to receive a portion of the proceeds--even if the sample's unrecognizable. Even getting permission doesn't always save you, as the Beastie Boys discovered. An article in a recent issue of SSA explores the issue in detail and concurs with Beck's assessement in 2005 that the legal issues with sampling will basically kill the practice in mainstream commercial music.
Covers generally require permission from the publisher, and some songs are never allowed to be covered--apparently, Bob Dylan's "Highway 61 Revisited" is one of them, although that hasn't stopped plenty of bar bands from trying.
But I was under the impression that parody is free game. There's legal precedent in a case called Campbell v. Acuff-Rose Music, in which rappers 2 Live Crew were sued by the publishers of "Oh Pretty Woman" for their parody cover version. The case went to the Supreme Court, and 2 Live Crew won. But even if they won, the risk of lawsuits (and plain old bad blood) is big enough that parodists are wise to ask permission--as Weird Al Yankovic does for every song he covers.
But the weirdest case I've ever heard is the story of Animal Collective, which interpolated a few lines from Stevie Wonder's "I Just Called to Say I Love You" in their song "Purple Bottle." According to somebody who worked on the album, Stevie Wonder's representatives would have given clearance if Animal Collective had sung the song more or less like the original. But because they changed the lyrics--"I just called to say I like you/I just called to wonder if you care"--and melody slightly, they thought Animal Collective were making fun of the song, and denied permission to release it. The band might have been able to go ahead anyway--it seems like it would be covered under parody law--but the legal fees aren't worth it for a band that sells tens of thousands rather than millions of records. So they re-recorded the song without the offending part.
Fortunately, some enterprising souls took it upon themselves to release 500 copies of the original verison on "white label" vinyl. As Animal Collective member Geologist explains, white labels are a common way that techno and hip-hop artists release music with uncleared samples--because there's no information about the artists or manufacturers on the label, there's nobody to sue. Plus, the releases are usually in such small volume that it wouldn't be worth the rights-owner's time to sue anyway.
It doesn't sound like parody to me, but rather a kind of sweet-hearted reinterpretation. If you want to hear for yourself, a streaming version is available here (at least today...it may be gone tomorrow). The borrowed lyrics start around 2:30.