You had originally posted to this thread:http://www.mytoos.com/forum/ubbthreads.php?ubb=showflat&Number=228487#Post228487
But I moved your post to its own topic in a different forum so you can get more responses.
The ABA forum is for one on one work on an individual problem -- which you may decide in time you'd like -- but for right now, more people can respond here.
If you do an advanced search, set the time frame for 10 years, and search on "screaming," you will find many posts as this is a very common problem. Perhaps some of the advice/experience on there may be helpful to you. Unfortunately, with birds, there is no step-by-step instruction manual. There is much observation and trial and error involved as each one is an individual. Of course, there are guidelines, but it boils down to finding something that works for your situation.
When faced with a problem like you list, it's really helpful to think about the situation and what your actions are teaching your too. Parrots study us and every time we interact with them, it's a learning experience. What does he learn when you do different things?
With screaming, it's helpful to try to get a handle on the circumstances surrounding the screaming. It may help to keep a journal so you can notice patterns. It's much easier to prevent problems than to react to them, and you may be able to do this if you can notice patterns.
What I did with my macaw was choose a word (sound, in my case) that he already made. For the first few weeks, whenever he made that sound, I dropped everything and ran to him. I was trying to teach him that acceptable sound = attention. I'd also give him attention when he was quiet, but I ALWAYS gave him attention for that noise, as I wanted him to learn how to communicate with me. (I.e. if he's lonely, he shouldn't scream, as that will no longer work, but he should make the sound.)
When it became clear that he understood sound = attention, I'd still always verbally acknowledge him when he made the sound (and still ignore the screaming) and still get to him, but I didn't drop everything as I had in the beginning when he was still making that connection.
Can you be a bit more specific about when he screams? Time of day? Is he hungry? You mention your kids -- where are they/what are they doing? Does he scream whenever he sees them (for example, if they're in the same room with him for 5 minutes, does he scream the entire time, or only when he first sees them?) What are you doing when he screams? Does he play with toys at all? Can he see you from his cage location? Etc -- stuff like that. Try to be specific about the environment that may exacerbate the screaming. It's much easier for you to change the environment and prevent screaming than it is to react to the screaming once it starts. (For example, let the machine get the phone and call them back when he's secured in his cage and you're in a room on the other side of the house.)
He's only 3, so he doesn't have decades of using screaming to get what he wants, which will make things easier for you. However, this isn't something that can be changed overnight. Additionally, happy toos scream on occasion, so you can't expect to completely eliminate it.