Preheat oven to 350 degrees F.
Arrange bread slices in 1 layer on 2 baking sheets and sprinkle half with Parmesan. Toast slices in upper and lower thirds of oven until crisp and golden, 8 to 12 minutes, and cool. Crostini may be made 2 weeks ahead and kept in an airtight container at room temperature.
In a colander or sieve drain tomatoes well, pressing gently on solids to squeeze out as much liquid as possible, and blot with paper towels. Chop tomatoes fine. In a bowl stir together tomatoes, oil, garlic paste, parsley, sugar, and salt and pepper, to taste.
Put 1 heaping teaspoon tomato mixture on each Parmesan crostini and top with 1/4 teaspoon olive paste. Spread each plain crostini with 1/2 teaspoon olive paste, and top with a heaping 1/2 teaspoon tomato mixture.
Garnish crostini with parsley leaves.