Yes — grapes are safe for parrots (unlike for dogs), but they're mostly sugar and water, so treat them as an occasional sweet, not a staple.
What it means for budgies
Keep grapes to a small treat for budgies — they don't need much sugar, and they're prone to obesity and fatty tumours. A budgie portion is tiny — a single bite or a little grated amount is plenty.
The full picture on grapes
The grape toxicity that worries dog owners has not been observed in parrots — grapes are safe for birds. The real issue is sugar: grapes are candy-sweet, and birds will fill up on them at the expense of vegetables and pellets. Raisins concentrate that sugar further.
Limit to one or two grapes, a few times a week.
Halve them for small species, and wash well — grapes are heavily sprayed.
Seeded varieties are fine; grape seeds are not toxic to parrots.