Is it permissible to assist in studies a sister who attends a mixed school and private lessons, and does not adhere to the lawful hijab, with the approval of the parents, even though the questioner is displeased with this situation?

It is permissible for you to help your sister explain the permissible sciences she studies at school, and you will be rewarded for this act if your intention is good. The fact that the school and private lessons are mixed, and that your sister does not observe hijab, does not prevent you from helping her understand beneficial academic subjects. This assistance is not aiding in forbidden acts, but rather aiding in a permissible matter. You are obligated to denounce forbidden acts according to your ability and in accordance with legitimate Islamic guidelines.