Know what the dust actually is
That fine grey-white powder filming the bars and the table under the cage is powder down. Cockatoos, cockatiels and African greys grow specialised down feathers that continuously disintegrate into this powder, and the bird spreads it through its plumage while preening to keep feathers waterproof and supple. It is a sign of a working feather system, not dirt or disease, which is why no amount of cleaning makes a powder-down species dust-free. Your goal is keeping the airborne and settled share low enough that the room, and everyone's lungs, stay comfortable.
Bathing moves dust off the bird first
The cheapest dust control happens on the bird itself. Most powder-down birds relish a shallow bath dish or a gentle misting from a spray bottle, and a bird that bathes two or three times a week carries far less loose powder into the air. Watch body language rather than forcing a schedule: a bird that leans into the spray, fluffs and shakes is enjoying it, while one that scrambles away should be offered a bath dish to choose for itself instead. Bath in the morning so feathers dry fully before the evening cool-down. Species differ in style here: many cockatoos plunge and soak with obvious delight, greys often prefer a fine mist drifting down from above like rain, and a reluctant bather of any species does better with a wet bunch of leafy greens hung in the cage to rub against than with repeated spraying it dodges.
Cleaning that traps dust instead of launching it
Dry dusting and feather dusters simply relocate powder down into the air you then breathe. Wipe bars, cage skirts and the shelf beneath with a damp cloth or damp paper towel, which lifts the film into the cloth. Change cage liner daily so droppings and dust do not dry together and puff up at the next change. Vacuum the cage corner with a hose attachment while the bird is in another room with the door shut, because vacuum exhaust is excellent at re-aerosolising exactly the fine particles you are trying to remove.
Air, placement and the purifier question
A true HEPA purifier sized for the room, placed near but not blowing directly on the cage, steadily strips powder from circulating air. Avoid any unit advertising ozone generation or an ioniser mode, both of which irritate avian airways far more sensitive than ours. Placement of the cage itself helps: away from fabric sofas and curtains that hoard dust, and off carpet where powder works into the pile. If anyone in the house has asthma or a known bird-related sensitivity, keep the bird's room as the low-textile, hard-floor room and wash hands after handling.
When dust stops being normal
Powder is normal; bald patches, broken feather shafts, constant scratching and a bird that looks unkempt despite preening are not, and they point to mites, skin trouble or illness that a vet should see. A sudden increase in dust alongside flaky skin on the feet or beak changes also deserves an avian vet visit rather than more cleaning. Note bath days, filter changes and who wiped down what in one shared place. Relatives you pick at home can be given access to that profile through KinsLe Family Sharing.