Heat the broiler to high, or your oven to 240 C / 475 F. Set an oven rack about 15 cm / 6 inches from the heat source.
Place the whole red peppers on a sheet pan 33 x 46 cm / 13 x 18 inch lined with foil.
Broil for 15 to 20 minutes, turning the peppers with tongs every 5 minutes, until the skins are blackened and blistered on all sides.
Transfer the hot peppers to a bowl and cover tightly with plastic wrap. Let them steam for 10 minutes, this loosens the skins for easy peeling.
Peel off the blackened skins, then pull out the stems and seeds. Don't rinse the peppers under water, this washes away the smoky flavor.
Pat the peeled peppers dry with a paper towel and add them to a food processor along with the feta, Greek yogurt, garlic, olive oil, lemon juice, smoked paprika, and chili flakes.
Blend until mostly smooth, about 30 to 45 seconds, scraping down the sides once. Leave a little texture if you like a chunkier dip.
Add the salt and black pepper, then taste and adjust. Feta is salty, so add extra salt a pinch at a time.
Spoon the dip into a serving bowl. Drizzle with olive oil, sprinkle with chopped parsley and a dash more smoked paprika, and serve.