To make the tomato salad, put the cherry tomatoes and red onion in a bowl and drizzle over a little olive oil. Season with a little salt and pepper and mix well. Set aside at room temperature.
Heat the vegetable oil in a frying pan set over a medium heat. Pat the sea bass fillets dry with kitchen paper and season with salt and pepper. Add the fish to the hot pan, skin side down. With a fish slice, press down on the llets for the first couple of seconds to stop them from curling up. Allow to fry without moving the fish for 4 to 5 minutes, depending on how thick the fish is, until the llets are cooked three-quarters of the way through and the skin is nice and crispy. Turn over and cook for another 1 to 2 minutes, until completely cooked through.
Transfer the fish to warm plates, skin side up, and turn o the heat under the pan. Add the lemon juice, butter and mustard to the still-hot pan and swirl to bring the ingredients together. Add the parsley and capers and spoon the pan sauce over the fish. Serve with the cherry tomato and red onion salad on the side.