Is a wife obligated to obey her husband who prevents her from visiting her sick mother, who is in need of care, or is she permitted to help her mother, and is she allowed to seek a divorce for this reason?

A husband is not permitted to prevent his wife from visiting her parents without a valid reason, especially if they are in need of her care and service. If the husband insists on preventing her without a legitimate Shariah reason, he is committing a sin, and the wife is permitted to go out without his permission to serve her parents. The wife's service to her parents should be limited by not harming her husband's rights. The wife can agree with her siblings to provide a maid for her mother, or she can do so with her own money, or she can serve her mother in a way that does not conflict with her husband's rights. If the husband insists on preventing her, she is not at fault for seeking a divorce, because preventing her from visiting her sick mother is considered a great harm.