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.