In a large bowl, whisk together 2 tablespoons of olive oil, lemon juice, minced garlic, cumin, smoked paprika, turmeric, salt, and pepper.
Add the chicken thighs to the bowl and toss to coat completely. Cover and marinate in the refrigerator for at least 30 minutes, or up to 12 hours.
Preheat a grill or grill pan over medium-high heat. Brush the grates with the remaining 1 tablespoon of olive oil.
Grill the chicken for 6 to 8 minutes per side, or until cooked through and nicely charred. Remove from heat and let rest for 5 minutes.
While the chicken rests, mix the Greek yogurt with a pinch of salt and a squeeze of lemon juice to create the yogurt sauce.
Slice the rested chicken into thin strips.
Assemble the wraps by spreading yogurt sauce on the warm flatbread, adding the sliced chicken, and topping with halved tomatoes and fresh parsley. Roll up tightly and serve.