Whose Responsibility Is It to Look after Parents In Islam?
Question

We are a family of nine children; all of us are married and have families. Our father is deceased, and we have an elderly mother. We take turns in looking after her, and one of her children serves or helps her every day. One of our sisters has excused herself from taking care of our mother on the grounds that this is the responsibility of the sons, not of the daughters, because the woman’s duties towards her husband and children take precedence over her duties towards her parents, as stated in Islamic teachings, and a scholar has given her a Fatwa to that effect, as she claims. 
Is what our sister says correct? Is this what the teachings of Islam say? 
Please note that she works and spends most of the day outside the home, and sometimes she travels for several days at a time, and she would not ask to be excused from travelling on the grounds that she has to take care of her husband and children.
Praise be to Allah.If the parents need to be looked after, it is obligatory for all their children, whether male or female, to help them, either by doing this themselves or by hiring someone to do it.
As-Safarini (may Allah have mercy on him) said in Ghidha’ Al-Albab (1/390): “Parents’ rights also include: serving or helping them if both of them, or one of them, needs that.”
In Al-Mawsu`ah Al-Fiqhiyyah (19/39) it says: 
“With regard to the child serving or helping his or her parent, or the parent asking his or her child to help him or her, that is permissible and there is no difference of scholarly opinion concerning that. In fact, this comes under the heading of honouring one's parents that is enjoined in Islamic teachings, and it is obligatory for the child to take care of his or her parent, or to hire someone to do that, when needed. Hence it is not permissible for the child to charge a fee for doing that, because it is duty that is required of him or her, and in the case of one who does a duty that is required of him or her towards someone else, it is not permissible to charge a fee for this.” 
Therefore, this service and help is obligatory for all of you, but if a woman's husband prevents her from serving and helping her mother , then the husband's right takes precedence , and she can make up for that by hiring a servant or by contributing to wages for a servant, if she has money. But if her husband does not prevent her from doing that, then serving and helping her mother is obligatory for her as it is for her siblings.
It is not appropriate for the husband, if he is a man of decency and dignity, to prevent his wife from doing that , especially if she is only going to help her mother one or two days a week.
Moreover, if the mother needs a woman to take care of her, then the most appropriate people to do that are her daughters, because it is more appropriate for them to undertake that than anyone else, especially if it is not possible to hire a woman to do that, or if the mother does not like strangers (nonfamily members) to see any of her private affairs.
So you should advise your sister and explain to her the importance of honouring her mother, and the seriousness of defiant disobedience to her mother, and tell her that it is more appropriate for the daughter to take care of her mother than the son, because taking care of an elderly person may require seeing the `Awrah, and for a woman's daughter to see her `Awrah is less serious than her son seeing it.
And Allah knows best.