All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad
is His slave and Messenger.
We ask Allah, the Almighty and Majestic, to set right the situation between your parents, and we advise you in striving towards this; for reconciling between two disputing parties is among the greatest acts of nearness to Allah, and when it concerns one's parents, it is also part of dutifulness toward them. We advise seeking the help of those who hold standing and influence with them, along with supplicating for tawfīq (divine enablement of success).
The original ruling is that it is not permissible for a woman to leave her home without her husband's permission, and mere estrangement or mistreatment does not make it permissible for her to leave without his permission. However, the jurists have mentioned that she is permitted to leave for what is indispensable to her if she finds no one to bring it to her. Based on this, it is permissible for your mother to leave for what she needs, even if your father does not permit her, provided she finds no one who can spare her the need to go out — while observing the Sharia guidelines of proper covering (ḥijāb), modesty, and avoiding the means of fitnah.
What we have said regarding your mother's going out applies to you as well with regard to accompanying her, if this going out is safe and no harm is feared for you or for her from it.
As for two women riding with a driver, this is permissible if one is secure from fitnah (temptation), for the jurists have permitted a man being in seclusion (khalwah) with two trustworthy women.
Allah Knows best.