Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat. Heat about 4 inches of oil in a deep, heavy-bottomed pot over direct heat until it reaches 360 degrees F.
For the beer-battered fluke: Whisk together 3 cups of the instant flour with the cornstarch, seafood seasoning, baking powder, salt and baking soda in a large bowl. Whisk together the eggs and beer in a separate large bowl. Whisk the wet ingredients into the dry ingredients until there are no lumps. Let the batter rest for 5 minutes so the flour can bloom.
Meanwhile, make the tartar sauce: Whisk together the mayonnaise, mustard, garlic, jalapeno, lime zest and juice in a medium bowl. Season with salt and pepper. You can add a splash of the jalapeno pickling liquid if desired for extra flavor.
Dry the fish fillets with paper towels and season with salt and pepper on both sides. Working in batches, dredge each piece of fish in the batter. Using a fork, lift each fish fillet out of the batter, let the extra batter drip down and then carefully drop the battered fish into the oil. Do not overcrowd the pot. Fry until the batter is golden brown and crisp and the fish is cooked through, about 5 minutes. Remove to a paper towel-lined plate and season with sea salt.
Season the remaining 1/2 cup of instant flour with salt and pepper. Dredge the lemon slices and parsley in the seasoned flour, then fry until crisp, about 2 minutes. Remove to the paper towel-lined plate. Add the fried fish, lemons and parsley to a platter and serve immediately with the jalapeno tartar sauce for dipping.
(Alternatively, you can fry the fish, lemons and parsley in 360 degrees F oil over medium-high heat on a stovetop.)