Add yeast, sugar, and water to a medium-large bowl. Stir well. Set aside in a warm place for 5 to 10 minutes. The mixture should be foamy and bubbly at this point.

Whisk in Greek yogurt, olive oil, and salt. Add flour, parsley, and za'atar. Stir together with a wooden spoon until the dough starts to come together. Use your hands to continue to bring the mixture together, then turn out on a floured surface. Knead for about 3 to 4 minutes, until dough is smooth and no longer sticky.

Divide the dough into 10 equal pieces. Place on a plate sprinkled with flour. Add balls to the plate and sprinkle lightly with flour. Allow to rest for 15 minutes.

Preheat a medium nonstick skillet or cast iron pan over medium-low heat.

While the pan is getting hot, roll out a piece of dough into a 6 to 7-inch circle. Brush lightly with oil. Once the pan is hot, add the rolled out dough to the skillet, oil side down. Brush the top lightly with flour.

Cook for about 1 to 1 1/2 minutes, until the top forms bubbles and the bottom gets golden brown in spots and around the edges. Flip and cook for another 1 to 1 1/2 minutes, until the bottom is golden brown.

Remove from heat and continue to roll, brush with oil, and cook with the remaining dough balls.

Sprinkle with additional freshly chopped parsley. Serve warm.

Once cooled to room temperature, store in a resealable bag at room temperature.