At what age do parents consider their children responsible?

It all basically depends on the parents. So parents find it suitable to see their child as a responsible young adult when they enter or about to be in middle school. However, many parents still see their children as their "babies" even though they are twenty. It also depends on the child. If you are trustworthy and haven't lost any toys or broken anything, then the parent will soon realize you matured a little and will see you as a responsible person. But then again, if you are one who argues and complains and make multiple mistakes, then your parent will need more time to see you as a responsible person. So, responsibility doesn't come in age but rather how mature and trustworthy you are.
Hope this helped :-)