Is expiation (kaffarah) obligatory for one who swore by Allah falsely for fear of a mother's anger?

Lying is forbidden. It can be avoided by using tawriya (equivocation) and ma'areedh (allusive speech). Lying is permissible to achieve a praiseworthy objective that cannot be attained through truthfulness. Pleasing one's parents is obligatory, and displeasing them is forbidden, as the Prophet, peace and blessings be upon him, said: "The Lord's pleasure is in the pleasure of the parents, and His wrath is in their wrath." You are not required to offer expiation, but you must seek forgiveness and repent for swearing to something untrue, and use tawriya in the future.