Bring a large pot of salted water to a boil. Cut the celery stalks into 2-inch pieces (reserve the leaves). Boil the celery until tender, about 10 minutes. Drain and rinse with cold water, then spin in a salad spinner and pat dry. Pulse the celery in a food processor until chopped. Squeeze handfuls of the puree in a kitchen towel to remove excess water.
Heat 1/4 cup olive oil in a large skillet over medium-high heat. Add the celery puree, 1/3 cup parsley, 2 teaspoons minced garlic and 1/2 teaspoon each salt and pepper. Cook, stirring, until soft, about 8 minutes. Transfer to a bowl and let cool. Reserve the skillet.
Mix the eggs, breadcrumbs and parmesan into the celery mixture. Roll into about 16 small balls; refrigerate until firm, 20 to 30 minutes.
Heat the remaining 1/4 cup olive oil in the skillet over medium-high heat. Add the celery balls and cook, turning, until golden brown, 8 to 10 minutes. Transfer to a paper towel-lined plate to drain. Pour off half of the oil from the skillet. Add the tomato paste, the remaining 1 teaspoon minced garlic and the red pepper flakes; cook over medium heat until the garlic starts to brown, about 1 minute. Add 2/3 cup water and whisk until emulsified, about 30 seconds. Return the celery balls to the sauce and heat through. Add the reserved celery leaves and the remaining 2 tablespoons parsley.