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, Gregor´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 Gregor´s transformation and subsequent alienation from his family, most notably, his father and mother.