In a large pan, heat olive oil over medium heat. Add the onions and cook, stirring occasionally, until softened, about 3 minutes. Stir in the garlic and cook until fragrant, 30 seconds. Add the ground beef, breaking up into small pieces, and stir until browned and cooked through. Stir in the parsley, salt, pepper, and paprika. Cook, stirring often, for an additional 2 minutes. Remove from heat.
In a medium bowl, whisk together yogurt, olive oil, and egg. Brush a thin layer over the bottom of a 9x13 inch baking dish.
Preheat oven to 350 degrees F and place rack in middle of oven.
Lay phyllo sheets on work surface and cover with a wet cloth. Place one sheet of phyllo in the prepared baking dish. Gently brush with the yogurt mixture. Repeat with half of the phyllo sheets.
Spread the cooked ground beef over the prepared phyllo sheets in baking dish. Place a phyllo sheet over the layer of ground beef. Brush with yogurt mixture. Repeat with remaining phyllo sheets. Brush the top with more yogurt mixture.
Bake in preheated oven for 40 minutes. Let sit 10 minutes before cutting and serving.