When my mother became ill (and she lived in another state), this is how I handled it.
I talked to my boss immediately to tell him there was a family emergency and what it was and that I would be going home on weekends for the forseable future and that I could get suddenly called away at anytime. And we worked out a way for me to make up hours so I could come back on Monday morning instead of up there and Sat and back on Sunday. I went into the discussion with a list of my current projects and where they stood and recommended who could take over for me when my mother died and I had to leave for the funeral (we knew her disease was terminal).
I made sure that nothing was ever left unchecked into source control when I left for the day (your boss could possibly give you a separate branch to check in unfinished work if need be). I also made sure the designated alternates for my projects were given copies of any emails they needed as well as being attached to the project in the PM system so they could read the discussions if they needed to.
As a new person this is a bit harder, but most people will understand about a family emergency. Since you are just developing your relationship with this manager and since there were some failures before you, it will give him more confidence about the situation if you talk to him right away and make sure you present to him how you will be handling making sure that whatever you are working on can be picked up by someone one else if you have to leave suddenly. I know we sometimes want to keep the "what is wrong" private, but it really is better if you tell your boss exactly what the problem is so he understands the severity of it. You can ask him to keep the information private from anyone except people who have to know like HR and his managers.
Now comes a really hard part, if you get one of those sudden, "come now" calls, take a few minutes before you leave to make sure he knows what is happening and where everything is, so he can assign it out. Five minutes of saving stuff to source control and informing him will help even though I know you will feel you have to leave this instant. Actually doing those things helped me to calm down enough to be able to drive.
Depending on how your dad is doing, you may need to take a long leave. Discuss this with your boss as well. Perhaps he will be open to you working remotely, so that you can be near your dad. Perhaps he will give you unpaid leave. You should plan on needing to take unpaid leave (since you are unlikely to have built any up)which means needing to control your finances starting immediately, so that you can. It is possible that he may be willing to work with you and get you some leave advanced. You won't know unless you ask, but be prepared for a no in this case.
By having an honest conversation right now, you and the boss can plan for the absences you may need to have and he will see you as a responsible person rather than someone skating off with an excuse. Of course, if he reacts badly to the news, this is someone that you will not want to work for in any event. Anyone who can't make accomodations for a family emergency without being a jerk about it is someone who is never going to be worth working for.
The days you work, you are likely to be less productive than before, espcially the first few days as you adjust to the new "normal" in your life or on any days when something changes. You might need to account for this in any time estimates you give. Be easy on your self and take the time you need to compose yourself when it gets to you by taking a walk or going into the rest room to have some private time. If you can keep enough control of yourself to keep producing good work, then you will impress your boss. But it will be hard, don't beat yourself up if you can't. Take the time you need so that when you are at work, you can focus as much as possible. You may need a day or two right away to go see your dad and for you personally to learn to cope before you return to work.
Do not let work and deadlines keep you from seeing to family, this is more important. I had a co-worker leave a week before a deadline that we had been working towards for a year because his wife had lung cancer (that was ten years ago and she is still alive BTW) and no one on the team resented it or thought he was awful for leaving. We would have thought it was wrong if he had stayed at work. People usually willingly pick up the work for someone else when the problem is this serious.
I am so sorry that you have to go through this and best of luck to yoour dad.