Adjust oven rack to middle position and preheat to 400°C. Line baking sheet with parchment paper. Roll out puff pastry to a 5 1/2- by 20-inch rectangle that's about 1/4-inch thick. Trim edges and cut into four rectangular pieces.
Place puff pastry squares on baking sheet and fold over edges to create a raised crust, giving a slightly curved shape to the edges pastry. Using a fork, prick bottom of the pastry.
Bake pastry until puffed, golden and crisp, 15 to 20 minutes. Let cool slightly, then gently press center of pastry shells to deflate it and create a well for the egg.
Sprinkle cheese onto center of pastry shells. Crack an egg into a small bowl and gently slide it into a puff pastry shell. Repeat with remaining eggs and shells. Arrange tomatoes, cut side up, on eggs and sprinkle lightly with salt.
Bake until egg white is just set and yolk is almost set, 8 to 10 minutes. Garnish with parsley leaves and serve immediately.