Puree together parsley leaves, green onions, half of the mint, the cumin, paprika, salt and pepper. Transfer to large bowl; beat in egg. Add beef and 2 tbsp water; mix well.
Shape beef mixture by heaping 2 tbsp into small sausage shapes; thread each lengthwise onto metal skewer.
Place skewers on grill over medium heat; close lid and grill, turning once, until no longer pink inside, 12 minutes (or broil, turning once, for 7 minutes).
Meanwhile, combine yogurt with remaining mint and minced parsley. Grill or broil pitas, turning once, until hot and lightly crisped, about 4 minutes; cut each into quarters. Serve koftas with pita quarters and yogurt.