Blend cherries (including pits) in a blender at low speed until skins have broken down enough to brightly color liquid (some of pits will be coarsely chopped). Pour through a sieve into a 4-quart pitcher, pressing on and discarding solids. Add lemon juice and sugar, to taste, stirring until sugar is dissolved.
Pour 3 tablespoons white rum, if using, into each of 16 tall (10-ounce) glasses filled with ice. Add 1/2 cup cherry lemonade to each and top off with sparkling water.
Cooks' note: Sour cherry lemonade can be made 1 day ahead and chilled, covered.