As was said, it is personal choice, and always best to be cautious. As for me though, I have 7 cats and 2 dogs- and one U2. I have had all my cats since they were babies, and they are now all between 11-14 years old. 5 of them have free roam, 2 live solely in my bedroom. Most of them were also exposed to a relative's cockatoo when they were younger- and the bird would chase the cats, NOT vice versa, lol. I *know* my cats, and although I know there is always risk, I feel confident in leaving my parrot alone on her top of the cage playstand if I have to run to the bathroom or kitchen or something quick, even though my cats are in the room. My cats have been on my lap while Lola (the U2) has been sitting with me- they pay her no mind whatsoever. One of my cats who lives in my bedroom (as he doesn't get along with a cat who is free in the rest of the house), has shown predator behavior towards her, such as too much interest in her in her cage one time when he was out. He will never be left alone with her, as I can tell there is a higher risk.
Same with my dogs- one, I raised from a puppy, and she wouldn't hurt a fly. She just wishes all the other pets would play with her, but knows they won't and won't bother trying. The other has a higher prey drive and although she has laid next to me with the bird on my lap, and other interactions, when the bird flies (Lola is clipped but she can fly a few feet and to the floor when she is startled) I see this dog get very excited- too excited to trust her completely. So they will not be left alone together. My cats have also always been indoor only- that can make a difference. They never learned to hunt anything. A few of them do enjoy when moths come in the house lol, but I've even had frogs loose on the floor or deck, or out in the yard (with me right there supervising of course), and they've shown little to no interest. Same with my dogs actually, they've met frogs and lizards and were taught that they were our friends, I know it sounds silly, but they do listen to me.
So I think there are a lot of variables- how well do you know your other animals, what are their prey drives like, how have they reacted to the bird... keeping a very close eye on any interactions in the beginning (and really forever) to see what level of interest the pet shows. Also keep in mind I'm talking about a U2- I would not have a smaller bird in my house as I do believe then it is just too much risk. And as for her reaction- she hissed and tried to bite my dogs at first, in only two weeks that behavior has stopped and, although I certainly don't trust that she won't bite them, she is learning that they are not going to harm her and reacting accordingly.
But as with anything, there are risks- we as parents have to judge the risks and take action as necessary. I know even my cats are at risk with two dogs in the house, but again, one dog was raised with the cats and just wishes they would play with her, the other came from a home with cats but we watch her quite carefully with them just to be sure. The only time she's shown any aggressive moves towards the cats were twice when they startled her from sleep on the couch as the cat jumped over her to get to the top of the couch. We made it clear that was very BAD, and she has since gotten used to cats jumping around and over her. I still know I can never trust her completely as I do my other dog, and I would be absolutely devastated if something happened to any of my cats because of her, but it is a risk we take having multi-pet households.