A crumb of hard cheese occasionally is tolerated, but birds are lactose intolerant and cheese is high in salt and fat — don't make it a habit.
Keeper's noteIf your bird is obsessed with cheese, use a rice-grain-sized crumb of sharp cheddar as a rare jackpot reward — the smell does most of the work anyway.
What to watch out for with cheese
Birds lack lactase, so dairy generally ferments uncomfortably in their gut. Aged hard cheeses (cheddar, parmesan) are very low in lactose, which is why a stolen crumb causes no drama — but they are dense in salt and fat, which parrot kidneys and livers handle poorly in quantity.
How to serve it
Hard, aged cheeses only, in crumb-sized amounts, rarely.
Never soft cheeses, cream cheese, or anything from the blue family.
Skip it entirely for birds on low-sodium diets or with kidney issues.