By referring to them as "his mother and his father" the reader can feel more close to the character. When saying "the mother and the father" one can relate less, feel distance between one and those parents. On the other hand, when saying "his mother and his father" the reader feels a little more close to the situation, feels deeper the relationship and can ever reflect him or herself in the character.
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.
