Is it permissible for children to work in permissible jobs to earn money, despite the opposition of their mother, who works in selling alcohol and believes her work is essential for their travels, fears for them from working, and threatens not to forgive them?

If both of you are capable of earning and the work adheres to Islamic regulations, then you must work. It is not permissible to abandon work and consume unlawful money. Obedience to the mother in quitting work is not obligatory if all the money is unlawful, because there is no obedience to a creature in disobedience to the Creator. However, if there is another lawful source, the money is considered mixed, and it is disliked to benefit from it for those who are not in need of it. In this case, obedience to the mother in quitting work is obligatory. It is advised to be gentle with the mother, try to convince her, and supplicate for her.