Cooked plain potato is fine in small amounts. Raw potato is not — it contains solanine and is hard for birds to digest.
Keeper's notePlain is the operative word: no butter, salt, milk, or frying. Fries and chips are a salt-and-fat problem on top of everything else. Sweet potato is the far more nutritious cousin — see its own entry.
What to watch out for with potato
Potatoes are nightshades: raw potato, and especially green or sprouting potato, contains solanine, which is toxic to birds. Cooking breaks down most of it and makes the starch digestible, so plain boiled or baked potato is a safe occasional food — though it's mostly starch with modest nutrition.
How to serve it
Cooked thoroughly, served plain and cooled.
Never raw, green-skinned, or sprouting potatoes.
Small portions — potato is filler, not nutrition, for a parrot.