Implications of quit claim or add to title for parents home

My parents are divorcing after 39 years of marriage in California. They jointly own a property (their house) that has tremendous value (est 4 million). Had they not decided to divorce, my sister and I would have directly inherited this home upon their death. However, now that my father has a new girlfriend, I am worried about how to make sure the asset (home) is protected from her or any other outsiders. My parents are the only ones on the title.

Here are my questions:
1) Should they add us to the title and then each quitclaim to my sister and me?
2) What are the tax implications of being added to the title or the quitclaim itself?
3)If we were to assume title, what would be the tax implications when my sister and I go to sell the house eventually?

We would personally like to get my father off the title and then let my mother live in in the house (which she wants too). We are also very interested in preventing the sale of the property because of their divorce - as we would rather they didn't liquidate the asset at this point and spend our inheritance away. Thanks for any help you can provide.