Some people say you should never own dogs and/or cats and parrots. I disagree, as you can see above, and to some degree, I also agree. Owning multiple species of animals adds a certain level of risk to your household. You yourself must consider the risk and whether or not it is worth it. Dogs and cats carry bacteria in their saliva that is harmful to parrots. In addition to that, dogs especially can crush a bird with a good bite (especially if the dog you have is, in my case, a pit bull :).
The other issue is that dogs and cats are predators, and birds are prey. Not only are birds prey, but they're interesting prey, because they fly away. Dogs and cats love to chase things that run away. If things don't run away, they're less fun, especially if the dogs and cats aren't hungry.
So, you've decided the risks are worth it, and you're going to have parrots and dogs and cats! What next?
1) The personalities of the parrot in question is as essential to making this work as the personalities of the cats and/or dogs. If you've got bird dogs, or your cats attack everything that enters the household, and constantly bring you back presents, reconsider a few times. If you're in love with (or own) a very timid, quiet African grey who's not very good with change or scary things, go look at Austin's pictures in this community, and think again. The best situation is to have some laid back dogs and/or cats, and a bird with enough personality and, well, obnoxiousness to stand up to anything. Your typical conure, Amazon, macaw -- good. The phobic grey? Bad. Also remember that size plays a serious role in how cats and dogs react to birds. Larger birds move differently. They don't flutter, they flap, and that's not as interesting. Also, be certain that your cats nails are trimmed. Keeping them short helps your furniture, helps your skin, and helps your birds.
2) Cage placement is essential. It is anyways, but here, it's even more important. Your birds must always be higher than the other animals. The other animals should NEVER be able to get on top of the cage. And the cage should be heavy enough and balanced enough that it cannot be tipped over by an impromptu wrestling match next to it. The cage should start out in a room where you can lock all the animals out unless you are there.
3) So, you have a sun conure, a big cage, and a room where the animals can be kept out. The cats are hovering outside the door dying to see what is inside, or the dog keeps barking every time the bird makes a noise. What do you do? Let the cats and the dog into the room. Let them look at the bird. Let the bird look at them. As goofy as this sounds, introduce them to each other. "Cat, this is bird. Bird, this is cat. Everyone, this is dog." Sit down and watch them interact for a while. If the cats attempt to do more than look, watch first. The cats left our bird alone after his first rushing attack inside his cage. I was about an inch away ready to intervene, but the cat decided that no, really, this scary running thing wasn't that interesting. If the bird reacts differently, or the cat is really insistent, squirt the cat with a squirt bottle.
Continue to allow the cats/dog to come in the room with the caged bird supervised whenever you can. Let them understand that this thing is staying around, and let them get over their interest. It took my cats about two weeks before they really didn't care at all about the birds any more.
During this time, if you have particularly bonded cats or dogs, spend a lot of one on one time with the cat or dog outside of the room with the bird. If you're in the room with the bird, let them in. If the cats or dogs do not ever see the bird except when it's out with you, they'll either get jealous or they'll think of it as dinner.
Once the cats or dogs get over the new addition, stop staring at it or approaching the cage so much and you feel comfortable with how they interact, leave the door open all the time. One thing to note is that if your cats indulge in 3 am kitty cat rodeo, be certain the cage isn't in a run path or that there are no neat toys near the cage. Cats slamming into a cage at 3 am is going to scare the bird. Leaving the door closed at night for a while isn't a bad idea.
4) NEVER LEAVE YOUR BIRD OUT OF ITS CAGE UNSUPERVISED AROUND CATS OR DOGS. That should really be phrased 'NEVER LEAVE YOUR BIRD OUT UNSUPERVISED.' Birds can get into trouble in a very short amount of time. Birds can fly off their stand and end up on the floor with the cats or dogs. Birds can eat power cords. Birds can get stuck in furniture. Birds can do a lot of bad things. If they aren't in a cage, be in the same room. One of the better stories I heard about why people shouldn't have cats and dogs was one about a woman with an African grey, who had the bird out on its stand, left the house, and the bird climbed down and was killed by the cat. This is not the cat's fault. This is the person's fault for not keeping an eye on her parrot.
Some people argue that birds in a house with cats and dogs should be flighted. I argue that they shouldn't be. Birds that fly around are much more likely to be pounced or snapped out of the air by a dog because flying means prey in a lot of animal's minds. The familiar bird who doesn't fly becomes something new when racing across the living room. And if the logic is 'well, if the bird ended up on the floor, how would it get away?' see #4. The other issue is that a flighted bird can get somewhere else FAST, far faster than you can intervene if need be.
To give you two examples of what not to do (and prove that I'm not perfect), our sun conure was learning how to fly. He was out on his stand in the same room as me, and I turned my back. Thirty seconds later, I heard a bird call, and figured it was our foster cockatiel in the other room. I looked over, no Tea. He had flown across the room, around the corner, and down the stairs, and was down looking lost. Our cats, as usual, didn't care in the least, but that could have been a tragedy, and I started paying more attention. Not enough, of course, to stop #2 from happening. Our grey fell off his stand when I was in another room, and by the time I came back because Tea was screaming, he had waddled across the living room and was trying to climb Tea's cage. Once again, cats didn't care. Still, it was completely against #4 and the safety comment, and a very bad thing on my part.
Safety Tip #2 -- Regardless of how much your bird loves your cats/dogs and vice versa, don't let them physically interact. This means no preening, especially of cats. Cats spread saliva all over their fur, as well as the bacteria that can be harmful. On top of that, if they're right next to each other, the bird can bite the cats/dogs, the cats/dogs can do something that you can't stop, etc. We often sit on the couch with a parrot and two cats, but they are always on opposite sides of us, and we're paying attention.
To summarize, no, I don't think everyone should have cats and dogs and birds. I don't think everyone should have birds either. :) It is very possible to live happily with all of them, and even have some amusing interactions. Our dog is thrilled to clean up the remains of pellets and veggies off the floor, and our kitten loves to sneak beneath the grate and steal pieces of kale from the bottom of the cage. Other than a continued danger to the dangling toys on their playstands (when the birds aren't on them), everyone gets along well. But I'm very lucky in personalities of all the animals involved. The dog was raised around cats and birds, and has a healthy respect for them. The cats are not hunters, and are generally happy to ignore anything 'prey-like' after a while. They ignored the hamster, they ignored the fish, so the odds were good they'd turn to ignoring the birds. And the birds that we 'trained' the cats with were a sun conure (who believes he is as big as a macaw) and two green cheeks (who believe they're as big as tigers), who were not at all scared, and after a few lunges and attempted bites, the cats decided they weren't worth it.
In fact, right now my main problem is that when I take our rehomed Amazon onto the couch for some scritch and foot toy time, the cats want to come and hang out too, and Buff has severe issues with them. I'm afraid the cats will get bitten! :)