Ripe tomato flesh is safe in small amounts, but tomatoes are acidic — and the plant's leaves, stems, and unripe fruit contain toxic solanine.
What it means for cockatiels
There's nothing specific to a cockatiel that changes the verdict on tomatoes — it's treated the same as for parrots generally. Offer a small, cockatiel-sized portion.
The full picture on tomatoes
Tomatoes are nightshades. The green parts — vines, leaves, stems, and green unripe fruit — contain solanine, which is toxic to birds. Ripe red flesh contains negligible solanine but is acidic enough to cause digestive upset if fed often.
Ripe, red flesh only — a slice or a cherry tomato half, occasionally.