Pour the 4 cups boiling water over the tea bags; cover and steep 5 minutes. Discard the tea bags.
Pour the tea over the blackberry mixture and let stand at room temperature for 1 hour. Pour the blackberry tea mixture through a wire-mesh strainer into a large bowl, discarding the solids, then pour the strained liquid into a large pitcher. Add the 1 cup cold water, stir and chill for 1 hour. Just before serving, add the sparkling lemonade.