First of all, as already mentiones: prevention! ake sure he has a big enough cage (Moustaches are very active birds that love to fly, i would recommend one that's at least 150*150*100cm, preferably bigger. A lot of natural branches, and i cannot stress enough the importance of daily foraging activites.
There's a post on enrichment in the toys section which i recently posted a reply to; go check it out.
First of all, it sounds like you have, probably not on purpose; rewarded him for screaming.
Parrots pick this up very quickly; they scream, someone tells them to "Shut up", gives them food so that they will stay quite, or maybe just walk in to the room withouth even thinking about that the parrot is screaming. In no time at all you have a parrot that screms his air sacs blue for attention.

To come running when they stop screaming is a very long process that most of the time is only somewhat effective, if effective at all. It is very hard to efectively reward "doing nothing".

Even if you only walk in to the room when he is screaming, by accident, once every week, that is still enough to keep the screaming behaviour reinforced.
Try to look at the differences between a slot machine and one of those machines that dispences soda when you put money in them. If you put money in one and it doesn't give you your reward (soda), you just get pissed and most of the time doesn't try again.
However, when it comes to slot machines, you try and you try and you try because you know that if you just do it a liiitle bit more, you'll get that money!
This is sort of the same deal. All it takes is for the behaviour like this to be rewarded once in a while for it to be effectively maintaned.
What you want to do to solve it is use differential reinforcement. You simply start rewarding some other behaviour with the same consequence your parrot is looking for when screaming. (i.e giving him attention)
I'm sure he has some cute noises that you guys would prefer over the screaming.
Simply make a habit of ignoring him and staying silent as death whenever he screams, and as soon as he makes cute noises, talk to him from the room you're in and go see him. It will be a lot of running and talking at first, but eventually this will work the same way as the screaming does today; it doesn't have to be reinforced all of the time.
Eventually, when the problem is solved, i would also cinsider getting another moustache to keep him company. Having two birds that get a long is a blast, and also so much better for the birds. They're really not suposed to be alone at all.

Good luck!