Stevens's father arrives to work at Darlington Hall at the same time as Miss Kenton. Obviously frail, he nevertheless insists on keeping up with all of the duties expected of a footman. Ultimately, he collapses in exhaustion and dies on the evening of an important international conference held at Darlington Hall. Oops.
Stevens totally admires his father and considers him a "great" butler. Stevens also relates a number of stories about his father that testify to his exemplary professional dignity. But their interactions remain weirdly impersonal and formal. Both of them seem so fully committed to their professional capacity as butlers that their relationship as father and son doesn't seem to matter.
Stevens continues to work the night his father dies:
"Miss Kenton, please don't think me unduly improper in not ascending to see my father in his deceased condition just at this moment. You see, I know my father would have wished me to carry on just now." (3.395)
This might seem heartless, but Stevens insists that his father would have wanted him to continue. And maybe he's right—Stevens Sr. represents an older generation of butlers who took their work seriously. But dying alone? That seems like a stretch. We hope that's a stretch.
Mr. Stevens also represents a possible destiny for Stevens himself: he may die a butler, in the middle of his butler duties. But unlike his father, Stevens may never even have a family to mourn his death.