If whole, clean the squid: pull the head from the body, remove the clear quill and innards, peel off the purplish skin, trim the tentacles just below the eyes and discard the beak, then rinse; slice the tubes into 0.5-inch rings, leave tentacles whole, and pat everything very dry with paper towels.
Finely chop the fresh parsley leaves; set aside.
Pour peanut oil into a deep, heavy pot to a depth of 2–3 inches and heat to 360°F, using a thermometer; set a wire rack over a sheet pan for draining.
Put all-purpose flour into a wide, shallow bowl.
Working in small handfuls, toss the rings and tentacles in the bowl to coat lightly, shake off excess, and slip into the hot oil; fry 60–90 seconds until curled, pale golden, and crisp.
Using a spider or slotted spoon, lift to the rack and immediately season with fine sea salt.
Allow the oil to return to 360°F between batches and repeat until all pieces are fried.
Pile onto a warm platter, sprinkle with the chopped herb, and serve at once with lemon wedges.