Answer:
The nuances of this question are too complex to address in a few lines. However, one could safely say, following the story and, especially, GregorĀ“s transformation into some sort of bug, that referring to the parents as the mother and the father, as opposed to his father and his mother, gives the reference a more impersonal tone, which is quite fitting with GregorĀ“s transformation and subsequent alienation from his family, most notably, his father and mother.