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.
What it means for budgies
Because budgies are prone to obesity and fatty tumours, treat cheese as an occasional, rationed treat rather than a regular food — a little goes a long way for a bird this size. A budgie portion is tiny — a single bite or a little grated amount is plenty.
The full picture on 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.
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.