A partner's dying parent is not an easy thing to cope with. During this stressful time, both you and your partner are going through many different emotions: sadness, anger and fear, to name a few. However, you are not losing a parent, your partner is. You need to be there to support, love and help them in any way that you can.
Tell your partner how sorry you are about their dying parent and that you will be there for them in any way you can. It may seem obvious, but it's important to lay a foundation of support and trust early in the grieving process.
Recognize that this is going to be stressful on both of you. Knowing that dealing with a partner's dying parent is going to be extremely difficult on everyone and probably one of the hardest things you will go through as a couple is an important step. Don't think this will be easy.
Figure out what your role is. If you know what it is that your partner wants you to do, you will have an easier time doing it. Your partner may want you with his family, or he may want you at home taking care of day-to-day things, such as laundry and bills. Be proactive about asking how best to help.
Continue living your life. Understand that dealing with a partner's dying parent is very emotional and demanding, but that you cannot just completely put your life on hold. Of course, your emotional involvement will depend on your level of commitment, and how well you know your partner's parent.
Listen to your partner. Do not tell her that you know what she is going through unless you do. Also, do not bring in your own personal anecdotes unless you think they will really help. She needs to know that you are listening to her and thinking about this situation. This is not about you.