Heat a large skillet or grill pan over medium heat and add the olive oil. Let it warm for about 60 seconds until it shimmers and is ready.

In a large bowl, combine the shredded chicken, mozzarella, cheddar, softened cream cheese, minced garlic, parsley, salt, and pepper. Mix well until the cream cheese is fully incorporated and no dry pockets remain.

Divide the filling into 4 equal portions. Lay each tortilla flat, spoon the filling toward the center, fold in the sides first, then roll the tortilla tightly away from you to form a sealed wrap.

Place each wrap seam-side down in the hot skillet. Cook for 2 to 3 minutes per side until deep golden brown and the cheese inside has melted fully. Do not increase the heat to rush this step.

Transfer to a cutting board, rest for one minute, slice each wrap in half on a diagonal, garnish with fresh parsley, and serve hot.