Combine first 3 ingredients in a large saucepan; bring to a boil. Cover, reduce heat, and simmer 10 minutes. Strain mixture through a sieve into a bowl, pressing rhubarb with the back of a spoon to remove as much liquid as possible. Discard solids. Cool to room temperature. Stir in juice. Cover and chill.
Stir in sparkling water just before serving. Garnish with lime slices.