In the beginning of the novel, Christopher's view of his father is rather innocent and immature. After his discovery about the death of the dog, however, his view of his father changes. The relationship is further damaged by the fact that Christopher discovers letters addressed to him from his mother, whom he'd been told was dead. This prompts him to leave his father and search out his mother. In the end, his father attempts to mend the relationship, and we are left to believe that they will mend their relationship.
As far as his mother is concerned, he's never really known her, but still she is ready to give up everything, including her country (she's living in London at the time) to move back to the US with Christopher. She seems very ready to step into her role. However, as Christopher has not had any experience with having a mother, their relationship builds slowly.
The Curious Incident of the Dog in the Night-Time