In a medium bowl, stir parsley, pistachios, garlic, lemon zest and juice, and ¼ tsp. salt. Stir in olive oil; set aside.
Season fish on both sides with remaining ½ tsp. salt. In a large skillet over medium-high heat, heat 1 ½ tsp. vegetable oil until very hot and shimmering. Swirl skillet to coat with oil. Carefully arrange 2 fish fillets skin side down in skillet and cook, undisturbed, until edges of skin are golden brown, 3 to 4 minutes. Flip and continue to cook until fish is just cooked through and underside is golden, 1 to 2 minutes. Transfer to a platter skin side up. Repeat with remaining vegetable oil and fish.
Top fish with pistachio gremolata if serving right away; otherwise, serve on the side. Drizzle with olive oil and sprinkle with sea salt, if using. Serve with lemon wedges alongside.
