Is the elder sister obligated to command her younger sister to learn matters of prayer and to perform household chores such as cooking, and is she sinful if she does not do so, given that the parents are alive and in good health?

As long as your sister is with her parents, she is a trust placed with them. They are responsible for guiding and raising her well, especially in teaching her religious matters, for Allah, the Almighty, says: "Allah instructs you concerning your children," and the Prophet, peace be upon him, said: "All of you are guardians and are responsible for your subjects." You are not obligated to do this except as part of enjoining good and forbidding evil, which is a duty upon every Muslim if they see an evil and there is no one else to address it. You should also help your parents in caring for and guiding her, because a sister can sometimes have a greater influence on her sister.