What changes the choosing answer
Plan for closed doors, secure cages, supervised out time, and a rule that predator pets do not get access to the bird. Training a dog or cat to be calm is helpful, but it is not a safety barrier.
The cage is not enough
A cage can be knocked, climbed, stared at, or pawed through. Place it where a cat cannot sit on top and a dog cannot jump, bark, or slam into the stand.
Out-of-cage time needs a room plan
Before the bird comes out, cats and dogs should be behind a closed door or otherwise securely away. Do not rely on someone promising to watch both at once.
Stress still counts
Even without contact, stalking, barking, staring, jumping, or repeated sniffing can keep a bird frightened. Chronic fear can affect sleep, appetite, trust, and behavior.
Safer default
If your home cannot reliably separate pets, do not bring a bird into that setup.