Preheat the oven to 400°.
In a large bowl, mix together ground beef, onion, yogurt, olive oil, oregano, garlic, salt, cumin, coriander, pepper, paprika and parsley.
Divide mixture into two equal-sized portions. Place one portion in the center of a 16x12-in. piece of parchment paper. Place another equal-sized piece of parchment paper on top; using your hands or a rolling pin, flatten into a thin rectangle, the same size as the parchment paper. Starting from the short end, fold the parchment over the meat, rolling it tightly into a 1-1/2-in. log. Repeat with the remaining portion.
Place beef logs on a rimmed 15x10x1-in. baking sheet. Bake 15 minutes. Drain off excess grease. Unroll the meat; break into smaller pieces. Return to the baking sheet. Broil 2-3 minutes or until the edges are crispy.
Serve gyro meat in flatbreads or pitas with shredded lettuce, tomato, onion and tzatziki sauce.