In a large mixing bowl, whisk together 1 cup all-purpose flour, 3/4 cup whole wheat flour, 3/4 teaspoon baking powder, and 1 teaspoon salt until evenly combined.
Add 1 cup plain whole milk yogurt to the dry ingredients. Stir with a spoon until the mixture forms a shaggy, cohesive dough.
Turn the dough onto a lightly floured surface (use a little of the measured flour for dusting) and knead about 5 minutes, until the dough is smooth and slightly elastic.
Cover the dough with a clean cloth and let it rest at room temperature for 10 minutes.
Divide the rested dough into 4 equal pieces. Shape each piece into a ball, then roll each ball into a thin circle on the lightly floured surface.
Heat a nonstick skillet over medium heat until hot. Place one rolled flatbread in the skillet and cook until the bottom is golden brown and the surface begins to puff, about 2–3 minutes.
Flip the flatbread and cook the second side until golden brown and cooked through, about 2–3 minutes more. Adjust the heat as needed so the flatbread browns without burning.
Keep cooked flatbreads covered with a clean cloth while you cook the remaining pieces. Sprinkle with freshly chopped parsley before serving, if using.