In a bowl, dissolve the yeast in the warm water with the sugar.
Let it sit for about 15 minutes, until foamy, to activate the yeast.
Add the olive oil and Greek yogurt to the yeast mixture.
In a separate bowl, mix the flour with the salt, then add it to the liquid ingredients along with the chopped parsley.
Mix with a wooden spoon until combined.
Turn the dough out onto a lightly floured work surface.
Knead for about 5 minutes, or until the dough is smooth and no longer sticky.
Add a little flour if needed, depending on the flour used. The dough should spring back when pressed with a finger.
Divide the dough into equal portions, lightly dust with flour, cover with a clean kitchen towel, and let rest for 15 minutes.
Heat a skillet over medium heat.
Take one ball of dough and roll it out into a disk about 15 to 17 cm (6 to 7 inches) in diameter, depending on your preference. Do not roll it too thick.
Lightly brush one side of the flatbread with olive oil and carefully place it in the hot skillet, oil-side down.
Brush the top with olive oil and cook for 1 to 2 minutes, until bubbles form on the surface and the bottom is golden.
Flip the flatbread and cook the other side for 1 to 1½ minutes. Do not overcook to keep the bread soft.
Transfer the cooked flatbread to a clean kitchen towel and continue cooking the remaining dough, stacking the breads as you go.
Once cooled, store the flatbreads in an airtight bag.