Kundera includes the story of his father's final illness to illustrate the absurdity and futility of human existence. Hey, you knew this was Eastern European literature, right?
There's no sugarcoating it, and Kundera won't even try. Things are bad—his formerly brilliant musicologist father loses the ability to speak at the end of his life. The irony of the situation overwhelms Kundera:
When Papa could speak normally, I had asked him very few questions. Now I wanted to make up for lost time. So we talked about music, but it was a strange conversation, between someone who knew nothing but a great many words and one who knew everything but not a single word. (VI.3.5)
There's nothing that Kundera can do to remedy the situation—he simply can't "make up for lost time." The strangeness of Papa's existence—knowing everything but not being able to articulate it—reflects Kundera's own disorienting experience in Czechoslovakia. Remember, his books were confiscated and then banned. It's just another piece of a bewildering backdrop.
The manner of Papa's death also grounds Kundera in the reality of life. It's not the beautiful "bluish" representation in Mann's story; it's undignified and meaningless work: "Death is terrible drudgery. My father lay dying for days with a fever, and I had the impression that he was working hard...he was concentrating like a rider on his horse trying to reach a far-off destination, but with no more than a final remnant of strength" (VI.12.9).
Papa is riding to that border where human desire and effort is laughable. Kundera can see no beauty in this—and not much hope, either. It's no surprise that he includes Papa's death story as a backdrop for Tamina's narrative. Her own ominous encounters with silence and meaninglessness emphasize, like Papa's aphasia (that's when you can't form words), the strangeness of a life without context.