What is the ruling on a daughter refusing to accompany her parents shopping because she wishes to stay home to listen to a religious lecture or read a book, and due to her unwillingness to go to mixed places where scantily clad women are present and music is played, in addition to her not having full شرعي (Islamically permissible) attire, knowing that her parents consider this disobedience?

Allah has made it obligatory to honor one's parents and obey them in what is right. However, there is no obedience to a creature in disobedience to the Creator. If the markets are as you described, with women's nakedness, mixing, and music, then it is forbidden to go to them except out of necessity. You are not obliged to obey anyone who commands you to go out to them, and you must not go out except in modest clothing that fulfills the conditions of legal hijab (definition). It is incumbent upon parents to protect their children and facilitate ways to safeguard their religion, without exposing them to temptations.