Perhaps there is truly no right or wrong. That said, you do have to look out for yourself. I'm sure you see the potential in him, alcoholics are not bad "demon people" or aliens but in active addition it can get very messy.
The bottom line here is that he will not stop drinking until he wants to and wants to do it for himself, for his life.
Nothing you do or say is going to change that. I don't know how long he has been drinking but i'm sure his parents have spoken to him about it - probably often. I'm sure they love him and he probably loves them too. Yet, that has not made one iota of difference. They bailed him out of jail, how many times have they done that? How many times have they been to the hospital?
4 days after leaving their house and going to yours: "They told me that they weren't surprised and didn't think it would take this many days for him to relapse" - they know him well.
My point is that it's not a good idea to think that he will look at you and at your relationship and go oh! I need to get sober for nitabug so we can fulfill that great life we both want! The thing is i'm sure he wants to be that sober, responsible Son to his parents and probably would like a nice normal relationship with a nice sober person (you), but that will not stop his drinking. He's proven that. He has to want it for himself.
He was in the hospital and his organs were failing. He continues to drink. That is very telling.
Alcoholism changes the brain. Craving for alcohol is all encompassing, a person has to be totally all in when trying to make that change, or they just keep drinking.
You cannot be responsible for his parents feelings. You cannot be responsible for him. How do you shake the guilt? I would ask why do you think you are responsible for him? You love him, you want him to be well but you are not responsible for him, he does get to make his own choices.
As for "fault" I wouldn't even go there really. I'm sure he didn't choose to become an alcoholic.