Sentences in Arabic can be categorized into two main types:
Nominal sentences: sentences starting with a noun, and they do not need a verb to complete the meaning (example: البيت كبير, Albaytu Kabeerun - The house "is" big).
Verbal sentences: Sentences starting with verbs and they do need a subject (donor of the verb, called Faa3il) and an object (called Maf3ool bihi, in accusative case).
in the above example, you can exchange the position of the verb (يسكن) with the subject (أبي) and the meaning will be the same still.
Correction: يسكن Yaskunu (he lives) [تسكن is Taskunu which means "she lives"].
Well, I would say the phrase given here is the common one actually. Some Arabic linguists suggest that the first word in the sentence is typically the word that the Arabic speaker unconsciously wants the listener to pay attention to or emphasize, and in that case it is "Abi" (my father). The two forms are equal in meaning and both are standard Arabic (have nothing to do with dialects really). But it might be more common to start it that way.