My father asks if it is permissible for him to do Hajj with my mother's money. He thinks that they share the money that she has earned from work. My mother traveled for work ten years ago while my father was caring for their four children until we became well-off.

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, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

It is permissible for your father to perform Hajj or ‘Umrah with your mother's money if she voluntarily gives it to him or permits him to do so. But taking from her money against her will is impermissible. A wife has the right of private ownership of her property. Your father's position of serving and caring for you instead of your mother does not allow him to transgress others' wealth. If there is an agreement between them or if it is the custom of people in the place where you live that he is entitled to a portion of the earned wealth in return for allowing her to work and replacing her in carrying out family duties, then he is allowed to get the agreed upon portion or what is customarily taken in such a case.

On the other hand, the wife's travelling for work while she has a husband or those (relatives) who can provide her with necessary means of living is against the sharia principle that demands that men to be in charge of women. It is against the woman's nature and her tasks, as well as the order of Allah for her to stay at home. This also exposes her to risks and evils that none know but Allah.

Allah Knows best.