In a large bowl, combine Greek yogurt, olive oil, lemon juice, minced garlic, ground cumin, ground coriander, paprika, turmeric, cinnamon, cayenne pepper, salt, and black pepper to create the marinade.
Add the chicken strips to the marinade and mix well to coat all pieces evenly. Cover and refrigerate for at least 1 hour, preferably 2 hours.
While the chicken marinates, prepare the garlic sauce by mixing Greek yogurt, minced garlic, lemon juice, salt, and olive oil in a small bowl. Stir until smooth and set aside.
Preheat a grill pan or skillet over medium-high heat. Lightly oil the pan to prevent sticking.
Cook the marinated chicken strips for 5-7 minutes on each side or until fully cooked and slightly charred. Remove from heat and let rest for 2 minutes.
Warm the pita breads briefly in a dry skillet or microwave until pliable.
To assemble each wrap, spread 2 tablespoons of garlic sauce evenly over the center of the pita bread.
Add a quarter of the cooked chicken strips on top of the sauce.
Layer with shredded lettuce, tomato slices, red onion slices, and sprinkle with chopped parsley.
Fold the sides of the pita over the filling and roll tightly to form a wrap.
Serve immediately and enjoy your Chicken Shawarma Wrap.