I have in the past dealt with sudden family emergencies with the attitude of "do what you need to do, fire me if you want. I have to go. Family is more important". Keep that in mind. You can get another job if you have to. You can't get another father, or get back time you needed to have spent with him.
I would talk to my boss in private and tell hinhim what is going on, and ask them how you can handle any time off needed, or needing to leave in the middle of the day if something happens.
Hopefully they will be understanding and work with you to accommodate you. That is how it has always worked out for me - bosses are humans with families of their own, so most can understand.
If your job and fathers health allows it, offer to work from wherever you are staying, or make up the time later.