What should I do to complete my wedding, given my mother's insistence that the girl's mother apologize first, even though my mother was the one who wronged her? And is it permissible to complete the marriage without the parents' consent?

Complete your marriage and do not hesitate. Try to gently convince your parents of the necessity of marriage, as it is a religious requirement and an urgent need, and that it is among the reasons for character formation and provision. Allah, the Exalted, said: "And marry the unmarried among you and the righteous among your male slaves and female slaves. If they should be poor, Allah will enrich them from His bounty, and Allah is all-Encompassing and Knowing." Strive to please them and be dutiful to them. And let your mother know that it is not permissible for her to attend if your father has sworn an oath to that effect.