Is it permissible for a young man to marry a girl despite his parents' refusal, which is not based on a legitimate Shar'i reason, and will there be a sin upon them for that?

There is no sin upon the girl if a man marries her without his mother's consent. The young man should make abundant supplications for his mother to agree and continue to involve intercessors. If this happens, then praise be to Allah. If it does not, the principle is that he should obey his mother because her obedience is obligatory, and his marriage to the girl is not obligatory. However, if the young man fears harm to himself by not marrying the girl, then it is permissible for him to marry her, and he should strive to please his mother afterward. As for the girl, if her marriage to him is not facilitated, she should turn her attention away from him and ask Allah to grant her a better husband.