This is a difficult question to answer because it is a subjective question for which the answer depends entirely upon your views of society, education, culture, and family roles. Some people might say, no, that Catherine has a family responsibility to sacrifice some normality in social and educational participation to help the family achieve its goals, such as allowing the mother to work. Other people might say, yes, that daughters and sisters are not servants in the home to fill in the gaps parents can't fill and that help isn't hired to fill. The answers reflect cultural and social standards, thus it is hard to say one is right and the other wrong.
In American culture, there is often an underlying belief, reinforced since the 1960s, that every member of a family has the right to strive for their goals and seek fulfillment and happiness. There is also an underlying belief that dates much further back that children in the home are meant to perform their fair share of household chores. In light of these beliefs, Catherine's mother is doing the right thing by working to (1) defray David's expenses; to (2) keep the family's standard of living at an acceptable level; to (3) fulfill her own needs for individuality and accomplishment. Further, in light of these, Catherine has a family obligation and responsibility to help take care of David even if it requires a sacrifice (or another sacrifice) on her part that makes her educational and social life sub-normal.
Other people might believe that it is the parents' obligation to shoulder all the responsibility for David's needs by hiring outside help or by reducing their overall standard of living--thus reducing costs and eliminating the need for Catherine's mother to work--in order to allow Catherine to be as free as possible to live as normal an educational and social life as possible. There are undoubtedly other belief systems, such as the Japanese one built on family duty, that might lead to different conclusions, but these two illustrate common difference in belief systems in America.
With opposing belief systems, each with justification within their own perceptions and normative structures, it is impossible to give a direct answer to your question. The best that can be done is to present some opposing belief systems to allow you to select a preferred system, then to answer the question for yourself from your own belief system.