Heat oil in a large skillet over medium heat. Add steak pieces, onions, parsley, salt, and pepper and cook until steak is browned on all sides, about 5 minutes. Remove from the heat.
Pat bread dough on a lightly floured surface into 4- to 5-inch rounds. Place about 2 tablespoons meat mixture on each dough round. Fold dough over the mixture to make a triangle. Pinch the edges of the dough together but make sure to leave the very center of the meat pie unsealed so steam can escape while cooking. Set on the prepared baking sheet. Repeat with remaining dough and meat mixture and cover with a clean kitchen towel. Let rise for 10 minutes.
Preheat the oven to 350 degrees F (175 degrees C).
Bake fatayer in the preheated oven until lightly browned, about 25 minutes.