Preheat a skillet or grill pan over medium heat and add olive oil. Warm each tortilla for 10 to 15 seconds in the microwave covered with a damp paper towel to prevent cracking when rolling.

In a mixing bowl, combine shredded chicken, mozzarella, cheddar, softened cream cheese, minced garlic, parsley, salt, and pepper. Add spinach if using. Mix until the cream cheese coats everything into a thick, cohesive filling.

Divide the filling evenly among the four tortillas, placing about half a cup in the center of each. Roll tightly, tucking in both sides as you go. Press the seam lightly to seal before placing in the pan.

Place wraps seam-side down in the hot skillet. Cook for 2 to 3 minutes per side without moving them until a deep golden crust forms and the cheese begins to ooze from the edges. Do not press down on the wraps.

Remove from skillet, slice each wrap diagonally in half, and serve immediately. Garnish with extra chopped parsley if desired.