Place ginger on several layers of damp cheesecloth. Gather edges of cheesecloth; squeeze cheesecloth bag over a small bowl to extract about 1/4 cup ginger juice. Discard pulp.
Combine ginger juice, orange juice, lemon juice, and sugar in a large pitcher; stir until sugar dissolves. Chill at least 2 hours. Stir in sparkling water; serve immediately.