Mince garlic and in a bowl stir together aïoli ingredients and salt and pepper to taste. (Aïoli may be made 2 days ahead and chilled, covered.)
In a bowl whisk together batter ingredients and pepper to taste. With a sharp knife peel celery root and cut crosswise into 1/4-inch-thick slices.
In a deep large heavy skillet (about 12 inches wide by 3 inches deep) heat 1 inch oil over moderately high heat until a deep-fat thermometer registers 375° F. Working in batches of 3, coat celery root slices in batter, letting excess drip off, and immediately press a parsley sprig into batter on one side of each slice. Carefully drop coated slices, parsley sides up, into oil and fry, turning them once, until golden brown, about 1 minute on each side. Transfer fritters as fried with tongs to brown paper or paper towels to drain. Return oil to 375° F before next batch.
