Prepare gremolata: In a small bowl, combine the parsley, horseradish, and lemon zest; set aside.
In a medium bowl, combine the flour, paprika, salt, pepper, and cayenne; set aside.
To cook the squid, heat 2 inches of oil to 375 degrees in a straight-sided pot. Be sure not to fill your pot more than halfway up the side with oil, because it will foam during frying.
Pat the squid dry, and toss in the seasoned flour mixture to coat. Let stand 1 minute. Shake the floured squid in a sieve to remove any excess flour. Fry in batches, without crowding, until golden brown, 1 to 1 1/2 minutes. Drain the squid on paper towels. (Frying a batch of squid will bring the temperature of the oil down. Be sure to reheat the oil in between batches and check the temperature again with a deep-fry thermometer before adding more squid.)
Season squid with salt and pepper. Sprinkle the gremolata over squid. Serve immediately with aioli and lemon wedges.