Add meat to a bowl and sprinkle with parsley, minced garlic, onion, cumin, coriander, smoked paprika, allspice, 1 teaspoon cayenne, 1 teaspoon salt, and black pepper. Use a fork or your hands to mix everything evenly. Cover and chill at least 1 hour and up to 12 hours.

For the sauce, add tahini, lemon juice, and water to a microwave-safe bowl and cook on High until very hot, about 30 seconds. Add to a bowl with yogurt, grated garlic, pinch cayenne, and 1/4 teaspoon salt. Whisk until smooth and light. Chill until needed.

Cut each pita in half and carefully open up without tearing the edges. Pitas can be warmed in a dry pan to soften them, which can make opening easier.

Divide meat into 4 equal portions, and transfer into each pita half. Press and squeeze pita gently to distribute meat in an even layer that comes all the way to the cut side.