Combine the yogurt, whole wheat flour, all-purpose flour, baking powder, 2 tablespoons chopped parsley and 1/2 teaspoon salt in a food processor and process until the dough is smooth. Wrap the dough in plastic and refrigerate for 30 minutes.
Preheat the grill to medium-high heat.
Divide the dough into 6 equal portions. On a floured surface, using your hands or a rolling pin, flatten each piece of dough into a large oval, about 6-inches in length. Oil the grill grate. In batches, grill the flatbreads until puffy and charred around the edges, 2 to 3 minutes per side. (Alternatively cook in a pizza oven or on a pizza stone for 4 to 5 minutes.) Hold the flatbreads on a plate under a warm towel as you cook the rest.
Combine the butter with the garlic and remaining 1 tablespoon minced parsley and 1 teaspoon salt. Brush the hot flatbreads with the butter mixture.