As long as the spacing between the bars is narrow enough to prevent injury if the bird tries to escape. The bird's head should not be able to fit between the bars. The door needs to be large enough to comfortably put your hand through, catch the bird, remove the bird, and replace the bird. For larger birds, it needs to be big enough so they don't rub feathers on the bars every time they turn around. And like people, birds like to stretch (I know an African Grey that does yoga). Just don't cramp the bird.
Here are some general guidelines:
SMALL: Examples, Finches, Canaries, Parrotlets, Parakeets, Lovebirds: Bird Cage Bar Spacing: No more than 5/8".Perch Diameter: 3/8" to 3/4"
MEDIUM: Examples: Cockatiels, Conures, Lories, Senegals. Bird Cage Bar Spacing: 1/2" to no more than 3/4"
Perch Diameter: 5/8" to 1 1/4"
LARGE: Examples: African Greys, Amazons, Macaws, Cockatoos. Bird Cage Bar Spacing: 3/4" to no more than 1 1/4". Perch Diameter 1" to 2"
Different species, different needs.
Small species (Finches, Canaries, Lovebirds, and Parakeets) deserve wider cages because birds travel side-to-side. Many of the small species never leave their cages. You have to let them fly because it's heart healthy.
Medium and larger birds have a different set of needs. So let's apply some common sense. Do you have a big bird? Are you away a lot? Then try to get a bigger cage. (Would you like to spend all day in a room the size of a closet?)
Bigger bird cages allow birds to move about much as we move about our home. Ideally parrots like three zones. The upper zone should have lots of toys and "cover". Birds feel safe up high, not easily seen, like in the wild. ( don't we feel comfy in our bedrooms?)
The middle zone is where they'll spend the day, playing with bird toys, eating, hanging out. The bottom is where they'll go to look for food and toys that may have dropped. Parrots are scavengers in nature.