My father says that it is not permissible for a wife to engage in even lawful trade even if she observes purdah.

Answer

Your father is correct. It is not permissible for a woman to participate in even a lawful activity without her husband's consent. If a woman intends to engage in even lawful activities in conflict with her husband's wishes, then she should not get married. She should remain a spinster and engage in an activity provided it is permissible in terms of the Shariah. We are sure that you understand that Nafl Saum (fasting) is a great act of ibaadat. But when the husband is home, it is not permissible for the wife to engage in Nafl fasting without her husband's consent. From this you will be able to understand that a married woman is not allowed by the Shariah to engage in even lawful activities and professions without the consent of her husband.