Is a marriage contracted under duress Islamically valid? And in this situation, would the children's lineage to their father be legitimate? And are the problems that the children suffer from related to the marriage being invalid?

In summary, the marriage that took place between the parents, motivated by the mother's fear of her father, is considered valid by the majority of jurists, such as the Malikis, Shafi'is, and Hanbalis. There is another view that considers marriage under duress impermissible, which is the madhhab of the Hanafis and the chosen opinion of Ibn Taymiyyah, Ibn al-Qayyim, Ibn Baz, and Ibn Uthaymeen. However, they deem such a marriage valid if the wife consents to it after the contract. Her consent can be indicated by allowing intimacy, demanding the dower and maintenance, or by continuing marital relations, having children, and not taking the matter to court.

Based on this, your parents' marriage is valid according to both opinions, and all its legal effects, such as dower, maintenance, and inheritance, apply. Their children are legitimate and are attributed to them. As for the problems faced by the children, they have no relation to the validity of this marriage, for every soul is accountable for what it has earned, and no bearer of burdens will bear the burden of another.