The operation was performed on the 13-year-old girl at a private clinic.
The parents filed a complaint with police.
"I want nothing but to hold the doctor accountable and to have justice for my daughter," the girl's mother, Hasanat Naeem Fawzy, told al-Masry al-Youm.
Egypt's National Council For Women, condemning the incident, said female circumcision is a criminal act and called on the government to investigate, al-Arabiya reported.
The girl's father, Mohamed Ibrahim, told the Egyptian daily on Sunday their daughter, Suhair al-Bata'a, died Thursday.
"We left our daughter with the doctor and the nurse. Fifteen minutes later the nurse took my daughter out of the operation to a nearby room, along with three other girls who the doctor was circumcising. I waited half an hour hoping my daughter would wake up, but unfortunately, unlike the rest of the girls, she did not."
Ibrahim said the doctor took his daughter back to the operating room and then summoned an ambulance to transfer her out of the clinic.
The father said when he asked what had happened the doctor responded that he did not have the necessary medical equipment to treat the girl.
"When we reached Aga Hospital, they told us that the she was dead," Ibrahim said.
The girl's mother said doctors at the hospital told her their daughter had died from an overdose of anesthesia.
"The same doctor conducted a similar operation on her elder sister two years ago and villagers visit this doctor because he has a remedy for everything at low prices," Ibrahim said.