Check calamari for breaks and slice into 1/8- to 1/4-inch rings.
Heat oil in a deep-fryer or large saucepan to 400 degrees F (200 degrees C).
Place 2 cups flour in a bowl. Whisk milk and eggs together in a separate bowl. Place remaining 4 cups flour, cornstarch, paprika, salt, pepper, and cayenne in a third bowl; mix thoroughly.
Toss calamari rings in the plain flour. Move to the egg mixture and thoroughly coat. Move to the seasoned flour and coat fully. Move back to egg mixture if not thoroughly coated; coat with seasoned flour again.
Submerge floured calamari in the hot oil until golden, 3 to 4 minutes per batch. Lift out with a slotted spoon, letting oil drip off. Drain on paper towels. Place calamari in a small bowl; check seasoning. Add parsley and lemon wedge for garnish.