Darzi was born in Baghdad, Iraq to Armenian parents displaced by the 1915 Armenian genocide. His family had lived in Erzurum, Ottoman Empire (now Turkey). His paternal great-grandparents, Tatyos and Elbiz Shiroian, had one daughter and four sons. Elbiz and her daughter, Arevalous (Darzi's grandmother), were the only survivors of the genocide. They fled on foot to northern Iraq with the aid of a friend of Tatyos. Darzi's father was born in Mosul and his mother was born in Baghdad.