Petitioning to leave PhD program with terminal masters due to illness in family - how should I expect my request to be handled?
I would do the following: 


Talk to supervisor: First of all, you need to talk to your supervisor about this, and see how much leave you can get. Very unfortunately, I see this every year many students have the similar situation.
Aim for 6 months leave first: You shouldn't just drop the whole thing. I would suggest you to take a leave for 6 month, and then extend it if necessary. 
Don't Lose Touch: I know it is a hard time, but do not just drop the whole thing completely. You can still read about the subject you are working on. Through university's VPN, you can still read research publications and be up to date.


Little Note on Big Picture: We all go through these sort of issues somewhere in our life time. We shouldn't make career suicide, or get into hard medication through drugs because one of our family members is passing away. We need to be strong, somewhere in our life time our parents pass away; and even though it is sad and hearth breaking we need to carry on and keep fighting.