Should one continue an engagement with a girl whom the parents initially approved, then later rejected without a clear reason, after the relationship grew strong and everything was agreed upon?

If the parents who decided to annul the engagement are the young man's parents, he must obey them. If they are the young woman's parents, they have the right to annul the engagement because it is not a binding contract, especially if there is a reason for it. The young man can seek help from good people to convince them to change their decision; otherwise, he should look for someone else and submit his affair to Allah, for perhaps what Allah has chosen is best.