What is responsibility, and where does it begin and end in the context of the parent-child relationship?
Responsibility means that an individual has a duty to another person. Responsibility goes hand-in-glove with accountability, or the taking ownership of one's actions.
A parent's responsibility, of course, begins at the moment of conception and continues at least through the teen years. A child's responsibility to his parents begins at birth and continues to adulthood, and often reintegrates as he and his parents mature.
Children go through many stages: infancy, toddlerhood, childhood, the teen years, and finally a transition to adulthood. Initially a child has no responsibility toward his parents, other than obedience, but he begins to test this at about age two; when he realizes that he cannot prevail, he progresses through childhood and again, at about age twelve, begins to assess and assert his relationship to his parents.
Still, his focus is himself, his friends, his interests, and while he does consider the feelings of others, this is often outweighed by his own desires. His accountability and responsibility to parental or authority figures are secondary.
When a "child" turns eighteen, in most jurisdictions he is legally an adult. At this point, however, many "children" are still evolving and becoming adults. Some, of course, take longer to fully transition and for many this will not occur until much later - often, the parents are to blame by continuing to enable dependence.
A child's responsibility to his parents, upon reaching this transitional stage, is quite limited. This is a time to focus on his own needs and his own chosen path through life; this is a time to learn, yet still have a soft place to land when necessary, and a time to work toward his own self-defined goals. Ideally, of course, this transition has begun with toddlerhood and is merely taking a larger leap during the high school years.
The parent-child relationship is just that, the affiliation and dependence of one upon the other; a child must depend upon his parents, but parents should not depend upon their children during those early stages. The transition to adulthood should include less dependence of the child on the parents and is, by definition, the process of attaining independence.
Children do, however, have a limited responsibility to their parents and will certainly exhibit such if the parent-child relationship is strong, healthy, and loving. Children will indeed communicate regularly with parents, and remain involved with and invested in these relationships, if they've been taught to value family and connection and communication.
They must, of course, learn to differentiate between independence and connection and not give in either temptation or pressure to blur that line of individualism. If the parent-child relationship is strong and healthy, this will usually not occur.
Many "children" today are known as the "sandwich generation": they are not only caring for and being responsible for their own offspring, but also attempting to be caretakers for their parents. These individuals must tread a fine line between maintaining relationships and exhibiting responsibility to their children, their parents, and themselves; they must also be aware of the pitfalls of reverting to dependence, while at the same time be willing to take on the responsibility of multiple generations.
Children do have a duty to become caretakers when parents are unable to manage their own affairs; especially children who have reaped the benefits of good parenting and continue to maintain loving relationships. Parents are the beginning of life for all children; some manage to promote the family, while others appear to do their best to destroy that unit.
As adults, children should be able to show some semblance of responsibility simply in gratitude for the very gift of life. The quality and quantity of their actions, care, concern, and loving, however, will and probably should be tempered with the type of relationship that was present throughout childhood.
Parents do have the majority of responsibility in the parent-child relationship; as the adults, they have the authority and the accountability to raise their children. When the children themselves reach adulthood, the tables are certainly turned.