What are the similarities and differences of the Parent-Child Relationships in the novel The Kite Runner?
I suppose you mean the difference between Amir's relationship to his father Baba and Hassan's (to the same):
Although the reader does not learn this until the end of the story, the critical difference is that of legitimacy. Being Baba's 'only' son, Amir benefits materially and socially from his father's wealth and prestige. He grows up in a household where there is considerable wealth and ease. He invites playmates over, receives expensive presents, and his filial relationship to Baba is clear. However, Amir feels that his father is somehow disappointed in him by the fact that he is not the great athlete he would have liked him to have been. This is more Hassan's natural gift, and Amir is somewhat jealous about the complicity (especially over football) they share.
Although he lives under the same roof, Hassan's life is quite different. Nobody knows (except Baba and Hassan's mother who left, and a friend) that Baba is indeed Hassan's father; to everyone, Hassan included, he is only the servant's son. Living conditions are adequate but much more modest; no expensive gifts are lavished upon Hassan, and when Amir receives his friends at home, Hassan must go off by himself. Hassan quite naturally accepts this double code of conduct, not realizing his true relationship to Baba. Then one year as his birthday approaches, Baba offers to have Hassan's harelip fixed, an expensive operation. By this Baba dares to express (for once) his true attachment to Hassan and steps over social barriers to help his son. Hassan is less disfigured after surgery, but of course remains stigmitized in a family way.
It is not until the latter half of the story that this relationship is disclosed to Amir. It is too late for Amir to "fix" his relationship with Hassan (Hassan being dead), but he has a new understanding of his father - not so perfect, but more human than ever. Baba never manages to come out and speak about this family secret with Amir, but he makes up for it somewhat when they start life together over again in California. Then Baba learns he has cancer. He is especially attentive to help Amir win Soraya by giving his paternal consent while there is still time.
An interesting comparison/contrast can be made between Hassan's relationship with Ali and Amir's relationship with Baba. Although Ali is not Hassan's biological father, he is his father in life, fulfilling that role in every way.
Ali and Baba, without wives, each raise their sons according to their social status in life and each man acts in good faith, believing his actions are in his son's best interests, but there the similarity ends.
Ali treats Hassan with love and acceptance. Having been tormented in his own childhood for his crippled appearance, he understands the stigma Hassan bears as the result of his facial deformity. After Hassan's birth, Ali's wife leaves him, but Ali never blames or punishes Hassan as the cause of his being abandoned. He chooses instead to love his son and do the best he can for him. When Hassan is falsely accused of being a thief in Baba's house, Ali leaves, taking his son with him. Ali would never abandon Hassan.
In contrast, Baba does not understand, accept, or value Amir's individuality when Amir is a child, and he does not understand, accept, or value Amir's choice of careers as a adult. He feels separate and estranged from his son through most of Amir's life. Baba tells his friend Rahim that if he had not witnessed Amir's birth, he would not believe Amir even was his son.