People today have largely lost the understanding that parody and pastiche are different but related formats.

Both are referential, but pastiches are affectionate and often deferential, while parodies poke fun in a biting, sarcastic way.

That's not really true, especially in music. For example, think of the term parody mass; it's not poking fun at all. It just employs previously-existing material as a part of a new composition.

Really? I've seen the terms "pastiche" and "parody" carefully delineated in scholarly works, though these have been commenting primarily in the classical eras and later, long after the parody masses. Charles Rosen (whose The Romantic Generation is now a classic, if controversial text) in particular has pointed out quite a few works he sees as pastiches as opposed to parodies.

Maybe I'm especially affected by earlier uses of the term, since I was most focused on parody in the sense of parody masses and musical borrowing in the Baroque. At any rate, I don't think the distinction is as simple as Rosen's usage might suggest (though I really enjoy Charles Rosen).

I once saw him perform a recital, including some Liszt. After the recital, I told him I thoroughly enjoyed The Romantic Generation.

His response? "That one has the best jokes."