Is it obligatory for a girl to visit her sick father, despite her mother's prohibition and threat to sever ties, and can she reconcile between her divorced parents?

You have done well by being eager to reconcile your parents, for this is one of the greatest acts of nearness to Allah and the best acts of obedience, especially between spouses, as they are a focus of Shaytan's attention. The most helpful thing for you in achieving reconciliation is to supplicate for them frequently at appropriate times. The hearts of the servants are between the fingers of the Most Merciful; He turns them as He wills. You can seek help from some trustworthy individuals known to them. Exercise wisdom in dealing with your parents, by maintaining ties with each of them and gaining their satisfaction, such as visiting your father without your mother's knowledge, and if you are unable to visit, then maintain ties with him through other possible means of connection.