In the mixer bowl, stir the water, sugar, and yeast. Let sit for 10 minutes.
Add the flour, yogurt, olive oil, and salt. Knead with the dough hook on medium until smooth and elastic, 8 to 10 minutes. Transfer to an oiled bowl, cover, and let rise until doubled, about 1 hour.
Mix the crumbled feta with shredded mozzarella until evenly combined.
Divide the risen dough into 8 pieces. Flatten each piece, fill with 2 tablespoons of cheese mixture, and pinch edges to seal. Place seam-side down, cover, and rest 20 minutes.
On a floured surface, roll each piece into a 6–7 inch/15–18 cm round.
Heat a skillet with a little olive oil. Cook one naan at a time, covered, 2 minutes per side, brushing with melted butter after flipping.
Sprinkle hot breads with parsley and garlic, and serve warm the same day.