Seriously, go see a councilor. I know you don't want to, but you should. It's perfectly normal to be depressed after losing your parents. It's nothing to be ashamed of. And that councilor may know of support groups made up of other people in a similar situation. It's going to be hard for you to figure out on your own. And why go it alone when you don't have to?
And don't worry about failing as a girlfriend. You have very few actual responsibilities to your boyfriend. Maybe you should take a time-out from that relationship, finish the grieving process, and then reconnect (if you want to).
If you believe in a higher power, that relationship can often provide some comfort. And if you know (as I do) that you will live again, and see them again, and can still have a family relationship, you can put this thing into a different perspective.
I'm sorry for your loss. I really am. Take some time, talk to someone who can help you get through your grieving process, and then figure out what you want to do with your life. My wife's mother is a psychologist, and she helps people work through their problems. Those who have a loss and don't grieve may look all together for a while, but at some point the facade will come down and the true loss will be felt. It's better to clean the wound so it can heal fully instead of festering.
Good luck. Don't do anything permanent right now, you're not thinking straight. And that's normal!