Yes — celery is safe, but its stringy fibers can cause crop impaction in smaller birds, so chop it or remove the strings first.
Keeper's noteChop finely or peel the strings off for small species. The leafy tops are string-free and most birds actually prefer them.
What to watch out for with celery
Celery is non-toxic, hydrating, and low calorie. The hazard is mechanical: those long cellulose strings don't break down and can tangle into a mass in the crop of a budgie or lovebird, which can require veterinary intervention.
How to serve it
De-string or finely chop for budgies, lovebirds, and parrotlets.
Bigger parrots handle whole sticks fine — great beak exercise.
Offer the leaves: safe, soft, and usually the favorite part.