This traditional Turkish dish dates back to before the Ottoman Empire and is a variation of a traditional Arabic dish, lahm bi'ajin (meat with dough). Another variation that might be more known internationally is the Gyro from Greece.
Heat the olive oil in a large skillet, add the onions and cook until they are soft. Add the spices and ground lamb, cook until the meat is evenly browned, breaking it up as it cooks.
Stir in the tomato paste, salt and parsley, cook for another minute. Set aside to cool. Can be made ahead.
When ready to cook, heat a griddle or large cast iron pan very hot. To four pitas, divide lamb mixture and top each pita with mixture, leaving a 1/2-inch border around the edge. One at a time, place on grill and heat for 5 minutes. Remove to plate.
Put 1/4 of the parsley, onions and tomatoes on the pita. Squeeze a lemon wedge over the top. Roll the pita up like a crepe. Wrap it in parchment to make it easier to eat or spear it with a toothpick to keep it closed.