For calamari: Soak the calamari pieces in the buttermilk in a bowl for at least 1 hour and up to 4 hours in the refrigerator.
For the lemon aioli: Add the mayonnaise, Dijon mustard, lemon zest and juice, salt and pepper to a medium bowl. Stir to combine and adjust the seasoning as needed. Set aside until ready to serve.
Preheat a deep-fryer or several inches of oil in a Dutch oven to 375 degrees F.
For the flour seasoning: Add the flour, paprika, salt and pepper to a large bowl. Drain the calamari and put into the flour mixture, making sure each piece gets coated with the flour. Dump all the contents into a colander and shake off any excess flour. Line a dish with paper towels and set aside.
In batches, add the lemon slices and calamari to the preheated oil and fry, mixing with a slotted spoon to prevent the calamari from clumping together, until golden brown, 2 to 3 minutes.
Right before you take the calamari and lemon out of the fryer, toss in the parsley leaves carefully; this will cook very fast and can splatter, so stand back. Take all the contents out of the fryer to the lined dish and season with salt.
Serve with the aioli, garnished with the zest and chopped parsley.