Arrests in such cases are rare because those responsible usually have powerful allies in the government, and the authorities are loath to challenge their armed bodyguards, who act as mini-militias.

No one has been arrested in the episode involving Mr. Hamidzai, although a police spokesman, Abdul Basir Mujahid, said an investigation was underway.

On Saturday, the children of another member of Parliament, Zaheer Sadat, a doctor from the Panjshir Valley with a history of brawling, got into a fight with the neighbor’s children. The adults in both houses opened fire on one another with automatic weapons, said Gen. Salem Almas, the head of the Kabul Police criminal investigation division. Five people were wounded, he said.

Dr. Sadat was not present at Saturday’s shooting, the police said. But in November 2015, when his convoy was stopped by the police, his bodyguards badly beat three of the officers, the police said. Dr. Sadat said that the officers had tried to attack him; the police said he was just angry at being stopped.

“No one is above the law,” said Mirdad Nejrabi, the head of Parliament’s Internal Defense Committee. “Being a member of Parliament does not mean one has immunity to do whatever he or she can to disrupt law and order.”

Just over a week ago, a brother-in-law and an associate of Mr. Hamidzai’s were stopped by the police while driving through the city drunk, according to a police official who spoke on the condition of anonymity for fear of being fired for discussing the case.

They beat up the officers and the police commander of the area who tried to intervene, the official said.