In a large bowl, combine minced garlic, plain yogurt, olive oil, ground cumin, ground paprika, ground coriander, ground turmeric, ground cinnamon, ground cayenne pepper, salt, and black pepper to create the marinade.
Add the thinly sliced chicken thighs 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 whisking together plain Greek yogurt, tahini, lemon juice, minced garlic, and salt to taste in a small bowl. Refrigerate until ready to use.
Heat a large skillet or grill pan over medium-high heat. Add the marinated chicken slices and cook for 6 to 8 minutes, stirring occasionally, until the chicken is cooked through and slightly charred.
Warm the pita breads in a dry skillet or microwave for 20 seconds until pliable.
To assemble each wrap, spread about 2 tablespoons of the garlic sauce onto the center of each pita bread.
Top the sauce with an equal portion of the cooked chicken, followed by shredded lettuce, tomato slices, red onion slices, and chopped parsley.
Fold the sides of the pita over the filling and roll tightly to form a wrap.
Serve immediately, optionally with extra garlic sauce on the side.