If your oven has a proof setting, preheat it to that; otherwise set the oven to its lowest heat (about 170°F) until warm, then turn it off.
In a medium bowl combine the active dry yeast, sugar, and lukewarm water; whisk lightly and let sit 5–10 minutes until foamy.
Add the vegetable oil, plain yogurt, and egg to the yeast mixture and whisk until combined.
In the bowl of a mixer (or a large mixing bowl), combine the all-purpose flour and salt. Add the wet mixture and mix until a soft dough forms and pulls away from the sides; add additional flour 1 tablespoon at a time only if the dough is too sticky.
Lightly oil a bowl, place the dough inside, cover with a towel or plastic wrap, and put the bowl in the warmed (now off) oven to rise until doubled, about 30–60 minutes.
Punch down the dough and divide into 8 equal pieces, shaping each into a ball.
Heat a 12-inch cast iron or other heavy skillet over medium-high heat and add about 1/2 teaspoon olive oil, spreading it to coat the surface.
Working one at a time, roll each dough ball to about 1/3 inch thick and roughly 5 inches in diameter, then place it on the hot skillet; cook until bubbles form and the underside is golden, about 1 minute, then flip and cook until the other side is golden.
Keep cooked naan covered with a clean towel to stay soft while you cook the rest.
Brush each hot naan with melted butter and sprinkle with chopped parsley before serving.