Kite Runner is the story of a sometimes dysfunctional relationship between a father and son. It focuses on relationships and forgiveness. A short story about fathers and sons that you might enjoy comparing to this novel would be Edith Wharton's His Father's Son.
This is a story about a father and son who also have a somewhat dysfunctional relationship based on a huge misunderstanding. In the end, the father and son realize they are more like each other than they had previously imagined, the same as in Kite Runner.
The father, Mason, is a self-made man who not only works hard to become a success so that his son, Ronald, can surpass him in life but who also then tries to live vicariously through his son.
At Harvard, Ronald had done exactly what the hypothetical Mason Grew would have done, had not his actual self, at the same age, been working his way up in old Slagden's button factory--the institution which was later to acquire fame, and even notoriety, as the birthplace of Grew's Secure Suspender Buckle. Afterward, at a period when the actual Grew had passed from the factory to the bookkeeper's desk, his invisible double had been reading law at Columbia--precisely again what Ronald did! But it was when the young man left the paths laid out for him by the parental hand, and cast himself boldly on the world, that his adventures began to bear the most astonishing resemblance to those of the unrealized Mason Grew.
Ronald becomes a success, very unlike his father. He is poised to marry a wealthy New York socialite and so his father tries to give him a large sum of money for his wedding. Ronald refuses to take the money because, he tells Mason, he has learned, from reading his deceased mother's letters that his father gave him many years ago, that he is not Mason's son, but the son of his mother's so-called lover, Fortune Dolbrowski. Mason explains to Ronald that it was he (Mason) that wrote the "love letters" to Dolbrowski, not Ronald's mother. It had all been a big joke. But because of the letters, Ronald learns that his father is actually a poet-at-heart, not the illiterate, self-made man he has thought.
There are parallels to Kite Runner because of the misunderstanding. In Kite Runner, Amir finds out many things he did not know about his father in similar fashion.
You can read the short story online at the link below. There is helpful information about Kite Runner here on enotes as well as information about Edith Wharton and her novel, Ethan Frome.