Line a colander or sieve with a double layer of cheesecloth that has been rinsed out and squeezed dry; allow cheesecloth to extend over edge of colander. Spoon yogurt into colander; fold edges of cheesecloth over to cover yogurt. Place colander in a bowl; refrigerate 12 hours. Scrape yogurt into a bowl, using a rubber spatula; discard liquid.
Beat yogurt and cream cheese at medium speed of an electric mixer until smooth. Press shredded cucumber and onion between paper towels to remove excess moisture. Add vegetables to yogurt mixture; stir well. Stir in salt and pepper. Set aside.
Cut 2 (1 3/4-inch) rounds out of each bread slice, using a cookie cutter. Spread 1 1/2 teaspoons yogurt mixture on one side of each of 16 white bread rounds; top each with a cucumber slice and a whole wheat round. Repeat procedure, spreading yogurt mixture on 16 whole wheat rounds, and topping each with a cucumber slice and a white bread round. Reserve remaining yogurt mixture.
Transfer sandwiches to a serving platter. Cover with plastic wrap; chill until ready to serve. Dollop remaining yogurt mixture on sandwiches. Garnish with parsley sprigs, if desired.