In a large bowl, combine the olive oil, garlic, cumin, smoked paprika, turmeric, coriander, cinnamon, cayenne, lemon juice, salt, and black pepper. Add the chicken thighs and toss until every piece is well coated. Cover and marinate in the refrigerator for at least 1 hour, or up to overnight for the best flavor.
While the chicken marinates, make the garlic sauce by whisking together the greek yogurt, tahini, lemon juice, garlic, and salt in a small bowl until smooth and creamy. Refrigerate until ready to serve.
Preheat the oven to 425 degrees F. Arrange the marinated chicken thighs on a parchment-lined baking sheet in a single layer with space between each piece.
Roast for 25 to 30 minutes, until the chicken is cooked through with charred edges and the internal temperature reaches 165 degrees F. Let the chicken rest for 5 minutes, then slice it into thin strips against the grain.
Warm the gluten free tortillas in a dry skillet over medium heat for about 30 seconds per side, or wrap them in damp paper towels and microwave for 20 seconds.
To assemble, spread a generous spoonful of garlic sauce down the center of each warm tortilla. Layer on the sliced shawarma chicken, shredded lettuce, diced tomato, cucumber, and pickled red onion. Garnish with fresh parsley, fold, and serve immediately.