Preheat oven to 400 degrees Fahrenheit (200 degrees Celsius) and line a baking sheet with parchment paper.
In a skillet over medium-high heat, cook ground beef and diced onion until meat is browned and onion is translucent.
Add minced garlic, salt, pepper, and oregano to the skillet and sauté for an additional 2 minutes.
Drain excess fat from the skillet and allow the meat mixture to cool for 5 minutes.
Stir the shredded mozzarella cheese into the meat mixture until evenly distributed.
Carefully slit the edge of each flatbread to create a pocket, or cut in half if using smaller pitas.
Stuff each flatbread pocket with approximately 3/4 cup of the meat and cheese mixture.
Brush the outside of each stuffed flatbread with olive oil and sprinkle with dried parsley.
Place the pockets on the prepared baking sheet and bake for 12 to 15 minutes or until the edges are golden brown and crisp.
Remove from oven and let rest for 3 minutes before serving.