What is the ruling on accompanying one's parents to public places and parks where immodesty and nudity are widespread for entertainment purposes, and is the son sinful if he goes with them, and how should a Muslim deal with this situation if his parents ask him to do so, fearing to be undutiful to them?

There is no harm in going out to public places like markets for necessities and errands, even with the presence of adornment and reprehensible acts, because need permits it. As for going out merely for recreation and leisure, it is not forbidden, provided one lowers their gaze and condemns evil. If parents command their children to accompany them to parks or markets, then it is obligatory to obey them.