Position the oven rack about 6 inches from the broiler and preheat your oven to broil (about 500°F / 260°C).
In a mixing bowl, combine the shredded cheddar, shredded mozzarella, sour cream, mayonnaise, and chopped parsley; season with salt and pepper and stir until evenly mixed.
Place the flat breads or halved French loaf pieces cut-side up on a baking sheet.
Divide the cheese mixture evenly and spread half onto each flat bread, smoothing into an even layer.
Broil for about 3–5 minutes, watching closely, until the cheese is melted and lightly golden at the edges.
Remove from the oven and let cool for a minute, then slice and serve warm.