From my understanding parody is not defamation.
That said, Weird Al hasn't actually distributed unmodified trademarks of other entities, to my knowledge. Even if that would generally be problematic, the context of a "featured partners" list as in TFA probably falls below a threshold of likelihood of confusion which arises in other unauthorized uses of trademarks.
Case law suggests that he is required to (and he does), license artist's music.
He's not parodying Miley Cyrus in 'Party in the CIA'. He's parodying the CIA. Because the lyrics and track aren't the subject, licensing of the track is required.
Similarly, If Repaer used a licensed font on the site, they'd have to license it.
I'm not sure about that. The definition of parody hinges on imitating an author or work (sometimes a whole genre), rather than on satirizing/critiquing subject matter unrelated to the author/work being imitated. He could write a song satirizing/critiquing the CIA and if it happens to imitate a song/style of Miley, then it's a parody of the latter, not of the former. Or a parody of nothing at all, in the strictest definition, since he's not satirizing/critiquing that which he's imitating.
When it comes to music copyright, certain aspects are copyrightable (therefore requiring license to use) and other aspects are not. Words and melody are (so Weird Al would need to license Miley's melody if he doesn't modify it sufficiently), but rhythm/chords/timbre/style/etc. are not (so Weird Al wouldn't need to license anything if he is merely copying those things from Miley). I think Al makes some songs with a copied melody requiring licensing, and some songs without that in which case no permission of any kind is legally necessary.
If you replace Smells Like Teen Spirit with Cliff Richards' "Saviours Day" it does not work, Cliff is not going to confuse and annoy your parents, his utterances aren't incomprehensible, and so on. The choice of song is necessary, which would justify protection.
My only nitpick is to remove the word "only" because there is at least one other possibility where he's "covered" in the sense of not legally needing permission: whenever his work falls short of copying the words and/or melody of the original song. If he modifies the melody enough (there's no specific threshold, but let's say he avoids a run of 4+ notes with intervals identical to the original work) and also doesn't copy lyrics or any other copyrightable aspects, then he's covered. It's very possible to achieve a song that makes your audience know exactly what you're going for, without copying any of the copyrightable aspects: you can take the chords, rhythms, instrumentation, accent, etc.
In the case of Smells Like Nirvana, he takes the copyrighted melody, which would require permission but for the exception you pointed out.
If he is printing such marks on his products, then it comes down to the "likelihood of confusion" test: will a consumer be misled to believe that the SKU Al is selling contains the official meat or silicon? That would be trademark infringement. In TFA, the natural gas trademarks are merely listed as "featured partners" so there's not much likelihood of confusion where a consumer would be misled into thinking that they'll get genuine natural gas from this unauthorized merchant, in my estimation.