As zovits already said in his answer, it is established by the main show canon that the Apple family origins are to be found in another region of Equestria. Sadly, the exact zone is still unknown, but it may be in some way related to Appleloosa (see S01E21).
That said, it is pretty clear that the creators did indeed want to portrait the family as a stereotypical family of farmers with a strong southern accent. The question would then be "why?".
Many episodes references the Apple family as one of the first families to settle in the land that would become Ponyville. This if fitting another stereotype. The Apples aren't just "farmers". They are colonizer, much like Grandma Elvira Duck family from the Disney comics was.
While this still not really answer the question, I think that the premise should be made to understand the rest.
As anyone reading probably knows, A.J. is the current bearer of the element of Honesty. While the show never actually explains what each element actually is in greater detail, we can safely assume that "Honesty" here isn't just a matter of "not telling lies". Multiple lessons indicate that one more meaning of "Honesty" is "to be honest to yourself", ie "be truthful to your origin, don't try to hide who you are just to please others". While it is not a direct confirmation or in-world explanation, it is pretty interesting to notice that in the first Equestria Girl comic special we get a glimpse of how the human world main six first meet. Here, you can see A.J. as she just moved to Canterlot High... And almost immediately she gets picked on for her southern accent and the way she dress (it is a bit weird, since she is also "picked on" by Babs Seed, who seem far older in this world than she is in Equestria). While at first she tries to change and "become like the rest of the students" (to the point of attempting to change the way she speaks), she will latter decide that she should be "true" to herself, and reverts to her old habits.
I know that the comics aren't really considered to be canon with the show, and an Equestria Girl related one probably isn't much better, but this seem to be as close as we may get to an official answer. The "Honesty" theme seem to be played on the entire family: be true to your roots, you traditions (the Apples seem to follow many family traditions and may even have some different ways to celebrate some events than the rest of Ponyville) - and so on. In this way, it seem only to be appropriate for them to still have the same accent their family "always had".
They still care for Apples in the same way. They still produce cider in the same way. They still celebrate with their relatives the same way... and they probably still speak the same way.