True, but that's just a reflective response to dissociate themselves from the current Islamic Republic of Iran.
After 1979 for many Americans Iran equated the Ayatollah, so saying you are "Persian" was a convenient way of dissociation.
I was in Europe at the time of the hostage crisis and met a Kurd from Iran who said he was from “Persia.” He was apologetic over the hostage outrage, but more concerned with the mistreatment of Kurds in Iran, naturally.