Season each fillet of branzino with salt and pepper.
In a bowl, whisk together the lemon juice, red wine vinegar, black peppercorns, minced shallots, garlic, olive oil, and parsley.
Preheat a grill to medium-high heat or grease a grill pan and place it over medium-high heat. Once hot, place the branzino fillets skin side up, grill for 2 minutes, flip, and cook for another 3 minutes.
Brush each branzino fillet with the lemon peppercorn sauce. Let the fish sit for a minute.
Remove the branzino from the grill and drizzle more with sauce.