blahblahblah, well done! Keep up your resolve, though. If you back down, they'll just be getting the message that "we can pull this stunt again whenever we want and we just have to apologise and blahblahblah will submit to our will again in only a few days." I'm sure you realise this, but it can be hard.
My parents used to do this to me, for nine freaking years after I moved out of home. And when I did call, all they would talk about was how much they hated me for not calling. It's pretty hard to dial a phone when you know that that is what's waiting for you on the other end. My many siblings (some older) still lived at home, and I think they still do, under their constant influence. It is very much a control thing. Even if you don't call eighteen times a day or whatever, they still control you by making you sweat over it in the meantime. Very manipulative and immature and massively disrespectful.
I hope your family situation does not go where mine did. My dad said a similar thing to "I hate you". His was, "I wish you'd died so you'd actually have an excuse <for not calling more frequently>, but it turns out you're just a horrible piece of <excrement>." After I gave them the cut in a similar way to you, they began pulling out the big guns. My father would suddenly turn up on our doorstep, apparently "in the neighbourhood" (about 8 hours out of the way), and have nowhere to stay. Cousins and relatives would suddenly be calling and emailling asking why I did various untrue and ridiculous things to my parents (stole money etc.--they may pull this on your brother, watch out.) Letters containing enormous guilt-trips and huge inconsistencies came in the dozens. I started getting 'emergency' calls at my workplace (parent hit by a car, bleeding to death, minutes to live, have to talk now before they're gone forever, etc.) which were all bogus. My innocent husband had a similar escalation of things. I really, really hope your parents see some sense, or at least that they aren't psychotic. But you seem to be handling things very well. Good luck to you!