Is it permissible to keep a marriage a secret and renew it later with the same wife, knowing that the father is ill and the family's positive view of the wife has changed, without their knowledge of the first marriage?

First: The husband is not obligated to divorce his wife if his father asks him to do so.

Second: The right of the father is great, and if the father has a heart condition, it is better that he not be informed of this marriage.

Third: You must ascertain the final position of your family. If you are certain of the sincerity of their position, then seek their permission for your marriage. If they give permission, that is what you desired. If permission is not granted, remain as you are so that their knowledge of your marriage does not lead to familial discord.

As for redoing the marriage contract, the first contract is valid if it fulfilled its conditions, so do not redo the contract lest it become a means for manipulation. You must strive to satisfy them by every means. If there is a genuine fear for your father's life, it might be said that redoing the contract is a matter of necessity.