Is my sister not Islamically obligated to serve her paralyzed mother, and is her husband entitled to prevent her from doing so, given the mother's verbal will that no one but her daughter should see her?

Serving one's parents when they are in need and infirm is obligatory upon all children, males and females, either by serving them themselves or by hiring someone to serve them. The sister must fulfill her share of serving her mother, especially in matters that require looking at awrah (private parts). If her husband prevents her, she is excused, and she must do what she can in terms of righteousness and kindness, or contribute to the costs of someone who serves the mother. The husband should not prevent his wife from serving her mother, but if it conflicts with his right, he may prevent her because the husband's right is more emphasized. There is no harm for male children in serving their mother in matters that require looking at awrah when there is a need.