Their job was to serve-as expressed by Jr.many times.I would think Sr's discussions out side of the staff around the diner table,were of the minimum-"Yes,Mi Lord"-"No,Mi Lord"-"As You Wish,Mi Lord".His accent wouldn't of been noticed by his employer.
Stevens Jr's more refined accent,would be a natural progression similar to the difference between a second generation Americans English as opposed to the first generation just learning to speak English.

3. Stevens Sr. entered the professional later in life and his natural accent was more resilient, more resistant to change.
4. His experience was in lesser homes, where standards were lower.
5. He was less invested in the profession than was his son, less convinced that adopting ways unnatural to his own was worthwhile. That would also partly explain how he ended up needing his son to give him a job - he's not only older, but less refined despite his years in service.
   I could see a simpler man scoffing at his son's determination to emulate toffs. In fact, the story he told at the servants' dinner table made fun of just such a creature.

Nothing to see here, move along.

It is mentioned that Stevens Sr had many years of butlering experience. But if that were the case then he would not have the unrefined accent he displayed in the film.
Butlers, learnt to temper their accent and put on a stilted refinement rather than the course cockney of Stevens Sr.
So there are two explanations:

1. Stevens Jr was lying (or at least exaggerating) when he said that Stevens Sr had many years butlering experience.
2. It was a goof.

good point!

I'm afraid I doubt that young Stevens would have had the chance to go to a public school, as private schools are called in Britain (far too expensive even today; the only chance would have been some sort of stipend).
I guess since Stevens grew up in a surrounding where he constantly heard the language of the upper-class, he was able to emulate it to a certain degree (but also note that the Doctor, who himself belongs to the middle-class, is not deceived by Stevens in the way the villagers are; he can still identify Stevens's class by his language; that's something you find in Britain even today).
It'd be interesting to analyse if (in the film and the novel) Stevens uses different levels of language when talking to his master and to other servants. Unfortunately English is not my mother tongue, so I don't recognize any differences.

That does explain a lot! And thanks for the speedy reply! Not only does your explanation explain a lot, but I also note that I think young Stevens was raised and educated within private schooling, whereas the old man must've been educated under public schooling.

In the novel Steven's very stilted, (pseudo-) upperclass language is part of his whole project of becoming the perfect butler (cf. also his explanation for reading the novel, which of course has to be taken with a pinch of salt). It is also necessary for the plot because it's due to his language (and his clothes and car) that the villagers think he might be an important man.
You can also interpret it as a part of his move away from his real, working-class father to his "ersatz"-father Lord Darlington. Or as a means not only for creating a certain, public persona, but also for hiding his feelings and true self.