In a shallow dish, whisk together the flour, Parmesan cheese, bread crumbs, paprika, garlic powder, salt, and pepper.
In another shallow dish, whisk together the eggs and milk.
Dip each onion ring first into the egg mixture, letting excess drip off, then dredge it in the flour mixture, ensuring it's well coated. Place coated rings on a plate.
Heat the vegetable oil in a large, heavy-bottomed pot or Dutch oven over medium-high heat to 375°F (190°C).
Carefully fry the onion rings in batches, without overcrowding the pot, for 2-3 minutes per side, until golden brown and crispy. Use a slotted spoon to remove them from the oil.
Drain the fried onion rings on a wire rack set over a baking sheet or on paper towels.
Serve immediately with your favorite dipping sauce.
Notes
For an extra cheesy flavor, you can add a little shredded cheddar cheese to the bread crumb mixture.