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You are right, I think the parents might have initially supported Mohammad Reza Pahlavi instead of the U.S. backed Shah. Check this out and perhaps it will make more sense,
In the twentieth century, Reza Shah renames this Persian territory Iran. He seeks to modernize Iran and engages the country with Western civilization. Oil is also been discovered in Iran which brings “another invasion.” Iran remains a neutral zone during World War II, and so the allied powers invade and occupy the country. They send Reza Shah into exile. He is succeeded by Mohammad Reza Pahlavi, “known simply as the Shah.” In 1951, a new ruler, Mohammed Mossadeq, institutes a number of reforms and takes back national control of the oil industry. The United States and Great Britain help to organize a coup against Mossadeq and he is taken out of power. The Shah returns to Iran and rules until 1979 and the Islamic Revolution.