Yes, taking care of your parents should be a central responsibility for each generation. Warehousing senior citizens in nursing homes robs them of dignity and puts the responsibility for taking care of family members on people outside the family. Instead, taking care of parents allows adults to pay their parents back for nurturing them as children.
Given that adults' parents helped to raise them into the people they are today, it would seem immoral to have adults simply ignore their parents as they age. As their parents age, their parents are able to do less on their own and need people to help take care of them; adults should be morally obligated to assist their parents in their later years, at least to some extent.
While adults should want to take care of their aging parents and ready to assume the responsibility if necessary, they should not be forced to do so. For some, the strain may just be too much from either a financial perspective. Others may simply be too overextended to commitments to make another. Some simply do not have healthy relationships with their parents.
No, adults should not be required to take care of their aging parents because those parents have had a lifetime to plan for the care they need in their old age. The parents should have saved adequately to handle their expected needs and should not be a burden on their children, who are trying to build lives of their own.