Place sieve over medium bowl. Line sieve with double layer of cheesecloth, leaving long overhang. Place Ricotta in prepared sieve and wrap cheesecloth around Ricotta, squeezing gently to release some liquid from the cheese. Cover and refrigerate until Ricotta has released most of its liquid and cheese is dry enough to form into balls. Discard drained liquid.
Line baking sheet with plastic wrap or parchment paper. Place drained Ricotta cheese in another medium bowl. Mix Pecorino Romano and Mozzarella into Ricotta. Season to taste with salt and pepper. Add 1 egg and stir to blend.
Place flour in small bowl. Whisk remaining 2 eggs in another small bowl. Place Parmesan in another small bowl.
Form Ricotta mixture into walnut-size balls (about 1 packed tablespoonful for each). Working with 1 cheese ball at a time, dip into beaten egg, then roll in flour, then coat with beaten egg again. Roll cheese ball in Parmesan, coating completely. Place on prepared baking sheet. Repeat with remaining cheese balls. Cover and chill at least 1 hour. Can be made 4 hours ahead.
Pour enough olive oil into large skillet to reach depth of 1½ inches. Lean deep-fry thermometer against side of skillet with bulb submerged in oil and heat to 360 degrees F. Prepare a plate lined with paper towels. Working in batches, lower a few cheese balls at a time into hot oil and fry until golden, turning occasionally, 30 seconds to 1 minute.
Using slotted spoon, transfer fritters to paper-towel-lined plate. Repeat with remaining cheese balls and serve with marinara dipping sauce, if desired.