Preheat a large skillet or grill pan over medium heat. Add one tablespoon of olive oil and swirl to coat the entire surface evenly.

In a medium mixing bowl, combine the shredded chicken, mozzarella cheese, cheddar cheese, softened cream cheese, minced garlic, and chopped parsley. Season with salt and pepper and mix until evenly combined.

Divide the filling into four equal portions. Spoon each portion onto the center of a low-carb tortilla, tuck in the sides, and roll up tightly to form a secure wrap.

Place each wrap seam side down in the hot skillet. Cook for 2 to 3 minutes without moving until the underside is golden brown.

Carefully flip each wrap and cook for another 2 to 3 minutes until the second side is golden and the cheese inside is fully melted.

Remove from the skillet, rest for one minute, then slice each wrap in half diagonally. Garnish with fresh parsley and serve hot.