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.
As adults, parents make choices. They make the choice to bring children into this world for the purpose of loving and caring for them. There comes a time when they must let go and let these children make their own lives and enter adulthood. At that point, they should plan ahead for their own end of life and make arrangements for themselves so they are not a burden to their children. They lived their lives, let your children live theirs. If you had children so they could take care of you in your old age then you are a selfish narcissist. If the adult child makes a decision to participate in the care of their elderly parents then it is their choice. But that is all it is. A choice. Not a responsibility. The only responsibility one has is to their children. The whole statement of "they took care of you so now you take care of them" is about the most ridiculous thing I have ever heard. They took care of you because they were responsible for you as a child. Changing the diapers of an infant or toddler is NOT the same as changing the diaper of an elderly adult body. It is not the same. The comparison is ludicrous.
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.