Season the branzino fillets with salt and a big pinch of pepper. Sprinkle with the turmeric.
Heat the grapeseed oil in a large pan over medium heat until shimmering. Add the fillets to the pan and cook for 2 minutes. Flip and add the butter, parsley and garlic to the pan. Continue cooking, basting the fish as the butter melts, until cooked through, about 2 minutes.
Transfer the fillets to a platter. Garnish on the side with the tomatoes and lemon wedges.