The discussion of rights between these two ethnic groups in Khaled Hosseini's The Kite Runner is a little tricky, because for the first part of the novel the differences are expressed in treatments, not rights. Once the Taliban moves into Afghanistan, the term rights is certainly applicable.
Before the Taliban arrives, the Pashtuns are the ruling class. They are the people with all the power, wealth, and position. Baba's family is a good example of how things worked for the Pashtuns and Hazara at this time.
Baba owns a large estate and was married to a Pashtun woman, so Amir is a full Pashtun--and he is well aware of it because Baba makes sure to make it clear. Baba's loyal and kind servant is Ali, who is a Hazara. Ali and his wife are both Hazara, as is their son, Hassan. While Hassan and Amir grow up together in a lovely way, they are not equals and they both know it. Amir has a bit of an epiphany when he reads in one of the books in his father's library.
For years, ...all I knew about the Hazaras [was] that they were Mogul descendants, and that they looked a little like Chinese people. School text books barely mentioned them and referred to their ancestry only in passing.
I read that my people, the Pashtuns, had persecuted and oppressed the Hazaras. It said the Hazaras had tried to rise against the Pashtuns in the nineteenth century, but the Pashtuns had "quelled them with unspeakable violence." The book said that my people had killed the Hazaras, driven them from their lands, burned their homes, and sold their women. The book said part of the reason Pashtuns had oppressed the Hazaras was that Pashtuns were Sunni Muslims, while Hazaras were Shi'a. The book said a lot of things I didn't know, things my teachers hadn't mentioned. Things Baba hadn't mentioned either. It also said some things I did know, like that people called Hazaras mice-eating, flat-nosed, load-carrying donkeys. I had heard some of the kids in the neighborhood yell those names to Hassan.
Rahim Khan, Baba's business partner and family friend, tells Amir about loving a Hazara girl when he was much younger.
"You should have seen the look on my father's face when I told him. My mother actually fainted. My sisters splashed her face with water. They fanned her and looked at me as if I had slit her throat. My brother Jalal actually went to fetch his hunting rifle before my father stopped him." Rahim Khan barked a bitter laughter. "It was Homaira and me against the world. And I'll tell you this, Amir jan: In the end, the world always wins. That's just the way of things."
So you see, it is less about actual rights than it is about history and practice.
Once the Taliban (which is comprised primarily of Pashtuns) takes over, the real persecution of the Hazaras begins. Assef, the bully who raped Hassan, is now a Taliban leader. He has always believed in ethnic cleansing (remember his admiration for Hitler) and hated the Hazaras, but now he has the power to act on his hatred. He routinely guns down innocent Hazaras without remorse and calls it "God's work."
"We left the bodies in the streets, and if their families tried to sneak out to drag them back into their homes, we'd shoot them too. We left them in the streets for days. We left them for the dogs. Dog meat for dogs."
Once the Taliban arrives, the Hazaras have no rights and live only at the mercy of the Pashtuns.
In the end, Amir is raising Hassan's son, so perhaps there is some hope for change.