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In brief: Probably. Celery is mostly fiber and water with little food content. It has been estimated that the calories expended in preparation, eating and digesting celeray are higher than the amount ingested by eating celery. Thus you can loss calories whilst filling yourself as long as you eat plain celery. Nothing untoward can come from eating celery.
Once you get your parakeet eating lettuce, move on to a few other vegetables, one at a time. I'd recommend celery next-parakeets love to crunch and gnaw things. Celery often has those little leaves on the stalk, and your keet already knows that leaves are tasty. So that is perfect. Try spinach-some parakeets like it, some don't. Move on to peas-they're round, rolly and fun to play with :) Give each new item at LEAST a week, to have your parakeet get used to this thing. Make sure each time that you show your keet that this item is edible. Go through the whole training process with each new item. Don't just leave the food in the cage and hope that your parakeet will realize it's food. Don't force your parakeet to eat only the new food, your parakeet might choose to starve rather than risk poisoning herself with something unknown.