“When your parent gets old enough to need assisted care, is it the responsibility of the child to give up their life and move back home and take care of them?”
Your wording in the question implies that you’ve already made up your mind, or that if you move back home you will resent your parents. I believe that the best way to handle these situations is to strike a compromise. You may want to have your parents move to you so that you can care for them. There are also some options for care which partially take the burden off the caretaking relatives.
About a year ago, I moved back home to live with relatives who have trouble getting around. It worked for both of us because I needed a cheap place to live and they needed help around the house, with errands, etc. There are certain things I can’t handle, but medicare covers home visits from nurses for my grandmother, who needs the help the most. They gently prod her into doing age-appropriate exercises, help her bathe, things like that.
If there are other relatives who might be able to help, you should talk to them about sharing the responsibility. Sometimes I feel unable to keep up with the demands, and I don’t want to be in a position where I resent my relatives. There is a saying, “many hands make light work.”