Your claim is for Promissory estopple (Detrimental Reliance). You entered into a contract and have so far, fulfilled your end of the bargain to your detriment relied on him holiding up his end of the bargain.
He can argue that the statute of frauds prevents this kind of contract, and a change in circumstances. It may be that one of a few solutions occurs.
1. The house is sold, a portion of the proceeds is set aside for the life-estate of your mother and father and split between them in the divorce. You get the rest.
2. You get a court ordered remainder interest and your father and mother get their life estate in the home. Which they can sell, rent or otherwise.
A practical solution would be for you to move into the home with your mother & son and you to pay rent for your father in his own apartment. Except for moving this leaves you in about the same position you currently are. To accompany this, your parents should deed the property to you and your mother as joint tenants with rights of survivorship, or as co-tenants with her cotenancy being a life estate and the remainder going to you.
Every legal matter is fact specific, and there are often nuances in every case. This is intended for comment only, and does not create an attorney client relationship.
I think you would have some problems with a breach of contract claim. Maybe you should look into a "quantum meruit" claim. Good luck!
P.S. You cannot give away anything when you are dead unless you executed a valid will before you died. And only things you own at your dead can be given away in the will. And you can change your will at anytime before you die. (If you don't have a will, you have no control over who gets your property.) The point is, when people promise to give you something when they die, you should be very skeptical.
I hope this information answers your question. If you need more information, simply add a comment or call me. Good luck!