Mix beef, onion, parsley, garlic, cumin, allspice, salt, and pepper. Halve pitas into semicircles; open pockets carefully.
Divide beef among eight pockets in even layers no thicker than 1/4 inch. Brush both outer faces with olive oil.
Heat two large skillets over medium heat. Cook four pockets at a time across skillets, 3 to 4 minutes per side, adjusting heat if bread darkens quickly. Repeat with remaining pockets.
Stand each pocket on its open meat edge 1 minute. Check beef centers reach 160°F; return underdone pockets to low heat another minute or two.
Mix yogurt and lemon juice. Cut crisp pockets into wedges; serve with sauce.