Preheat oven to 400 degrees. Place chicken breasts in a large baking dish. Spoon babaganoush on top of each chicken breast, so the top of each breast is evenly covered with a thick layer of babaganoush. Sprinkle top with paprika and with some of the parsley.

Bake for about 30 minutes, depending on the thickness of the chicken breast, until chicken is cooked through. Serve with additional chopped parsley as a garnish.