The word tasattur, which means the veiling of a woman, doesn't exist in the Qur'an. The Arabic expression humur (mentioned in the hadith) refers to a cover, not specifically a headscarf. The connotation of "humur" in the sense of a headscarf emerged much later, following the interpretation and practices of sectarian scholarship. The Qur'an speaks of the bosom as the part of the body that must be covered, and not the head. The words that correspond to the headscarf worn by Arabs are mýkna and nasýf, you can consult any Arabic dictionary to verify this.
Tell your parents to stick that in their pipe and smoke it.
P.S. I'm not a muslim.