Place sugar and 1/2 cup of water in a small saucepan over medium heat. Stir the mixture until the sugar dissolves. Set the simple syrup aside and allow it to cool.
Place 1/2 cup of water and dill in a blender and blend until the ingredients are thoroughly combined. Pour the mixture into a bowl through a fine mesh strainer and discard the pulp left in the strainer.
Add the simple syrup to the dill water. Add the zest and juice of the lemon to the mixture and stir.
To serve, pour one part dill mixture to two parts sparkling water in glasses over ice.