Combine water, sugar, and active yeast in a large bowl. Yeast should be foamy and bubbly within 5-10 minutes.
Add into the bowl the bowl with active yeast the Greek yogurt, olive oil, and salt and whisk to combine, then add in flour and parsley. Stir with a wooden spoon until the dough comes together. Turn dough out onto a well floured surface and knead until dough is no longer sticky and springs back when lightly pressed, about 3-4 minutes.
Divide dough into 10 equal pieces, sprinkle lightly with flour, and cover with a kitchen towel to rest for 15 minutes.
Preheat a medium nonstick pan over medium-low. While pan is heating,, roll one of the dough portions into a 7-inch circle. Brush top lightly with olive oil. Place disk oiled side down and cook until the top side is covered with bubbles and the underside is golden around the edges and in spots, 1½ minutes. Brush the topside lightly with olive oil and flip to cook another minute. Repeat rolling, oiling, and cooking with remaining dough portions.
Stack flatbreads and cover with a kitchen towel to keep war and pliable as they come out of the pan. When cooled, if not using right away, store in a ziplock bag. Reheat in a pan without oil for a minute on each side or heat in the microwave on low power until warm.