Is expiation obligatory upon the daughter who violated her father's oath by dyeing her hair without his knowledge, fearing his displeasure if he knew?

Obeying parents is obligatory in everything that is not an act of disobedience. Disobeying their command is an act of undutifulness ( 'uquq). If a father takes an oath, obeying him becomes obligatory, unless fulfilling his oath involves a sin or hardship. Therefore, your action is considered an act of disobedience and undutifulness, and you must seek forgiveness and repent. Regarding your father's breaking of his oath, if you do not anticipate any harm from informing him, then it is correct to inform him so that he may atone for his oath or permit you to do so. However, if you fear harm for him, then there is no blame upon you for not informing him, and it is permissible to follow those who say it is permissible for someone to atone on behalf of another without their permission.