What is the expiation for adultery, and how can marriage proceed despite parental refusal, after repentance from adultery that occurred between a young man and a young woman?

The expiation for what you have done is sincere repentance, remorse for the sin, seeking Allah's concealment, and not informing anyone of the sin, along with striving in obedience to Allah and increasing in good deeds. Try to convince your parents to agree to him in a gentle manner, while explaining your right to marry him if he is suitable. You can also seek the mediation of someone influential with them. Sever your relationship completely with this person until the matter is settled.