Welcome to the forum! There is something called a thunder shirt that people buy for dogs that are terrified of thunder - Amigo's owner should get him one because it will help. But it will not be enough by itself and I think that what you need to do is to desensitize the dog to the screams BUT this needs to be done very gradually and it requires a person being there all the time - and I don't know if this is doable in your case. If it is, you need to start by always keeping the dog in a different room with the door closed at all times between the two rooms as this will reduce the decibels. The other thing is to always have Amigo's owner with Amigo and, as soon as the bird screams, he/she needs to hug the dog and comfort him with the phrase he/she uses for this purpose. I have different phrases for my dogs, there is the generic "It's OK!" repeated several times in a firm but very calm tone of voice and usually followed by a "You are a good boy [or girl]" but I use other phrases for more specific situations and, in a case like this, I would use a specific 'bird phrase' - maybe something like I use for scary loud noises coming for the TV for which I say: "It's a movie! It's just a movie!".
The whole thing will take time and it will be very gradual but you should not have them in the same room until the dog is completely oblivious to Rio's screams. I have 8 dogs in my house right now and they don't react at all to the parrots screams - not even the cockatoo's and I can assure you that an ekkie's screams are nothing compared to a too's
Now, I hope you don't take this the wrong way because I don't want to offend you but I was actually quite upset when I read that poor Rio trembles in fear when he sees the dog or the cat... I LOVE birds, all birds, mine, yours and every single one out there and, sometimes, I am a bit tactless when I give my opinion to bird owners because I tend to put the birds wellbeing and happiness before their feelings so, please, bear with me on this and just think about what I am going to say: Ekkies are EXTREMELY fragile when it comes to their health - much more so than any other species of parrot -and this is no exaggeration! This is because they have a VERY specialized diet in the wild, a diet that we have not been able to reproduce in captivity so far [Laurella Desborough and all the early importers and breeders of eclectus did this species a HUGE disservice by making them part of the pet trade - so much so that she was called Cruella Desborough' in rescuers sites
] The very inadequate diet causes their liver and kidneys to suffer tremendously as well as their cardiovascular system to the point that they all die way before their time. A parrot that size should have an expected lifespan of around 50 years but there has not been a single ekkie that has lived more than 30 years and that was a wild caught kept in a zoo in its country of origin! In captivity, they all die in their twenties. Taking into consideration that stress depresses the immune system and that fear is the greatest stressor... well, you add it up.