In a large bowl, combine Greek yogurt, olive oil, minced garlic, ground cumin, paprika, turmeric, coriander, cinnamon, cayenne pepper, salt, and black pepper to create the marinade.
Add the chicken thighs to the marinade and coat thoroughly. Cover the bowl with plastic wrap and refrigerate for at least 1 hour, preferably 3 hours or overnight for best flavor.
Preheat a grill pan or skillet over medium-high heat.
Remove the chicken from the marinade and cook for 6-7 minutes on each side, or until the chicken is cooked through and has a nice char. Internal temperature should reach 165°F (74°C).
Transfer the cooked chicken to a cutting board and let it rest for 5 minutes. Then slice into thin strips.
Prepare the garlic sauce by mixing Greek yogurt, minced garlic, lemon juice, salt, and tahini in a small bowl until smooth. Set aside.
Warm the pita breads in a dry skillet or microwave for 20 seconds until pliable.
To assemble each wrap, spread about 2 tablespoons of garlic sauce onto the center of the pita bread.
Add a quarter of the sliced chicken, followed by shredded lettuce, tomato slices, red onion slices, and chopped parsley.
Drizzle an additional tablespoon of garlic sauce over the fillings if desired.
Fold the sides of the pita over the filling and roll tightly to form a wrap.
Serve immediately and enjoy your Chicken Shawarma Wrap.