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 African greys
In moderation African greys can enjoy cheese, but it's rich, so keep it to a treat rather than a daily food. Greys can take a larger portion than small parrots.
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.