Preheat the oven to 350F and line a 9 by 5-inch loaf pan with parchment paper.
Remove any bones from the fish with tweezers and cut the fillets in 1/2-inch pieces. Combine the fish, salt, and red pepper flakes in a food processor and process until finely pureed, working in short pulses. Add the egg whites and eggplant caviar and process until combined. Fold in the chopped parsley. Pour the mixture into the prepared loaf pan, making sure there are no pockets of air, and smooth out the surface with a spatula. Clean the bowl of your food processor thoroughly.
Set the loaf pan in a large baking dish and pour hot water into the dish so the bottom of the loaf pan is covered by an inch. Bake for 35 minutes, until the top of the mousseline is set.
While the mousseline is baking, prepare the coulis. Drain and seed the bell peppers and combine with the olive oil and garlic in the (cleaned) bowl of your food processor. Process until smooth and transfer into a small jug or serving cup.
Set a cooling rack over a plate. Remove the dish and pan from the oven. Lift the mousseline from the loaf pan by pulling cautiously on the parchment paper, and transfer onto the rack. If cooking juices leak from the loaf (this will or won't happen, depending on the eggplant caviar that you used), cut a few slits in the parchment paper at the base of the mousseline so the juices will drain down into the plate.
Let cool until slightly warm, or at room temperature. Cut in slices, garnish with parsley, and serve with the bell pepper coulis. (The mousseline and coulis can be made up to a day ahead: wrap the cooled mousseline tightly in foil, cover the coulis with plastic wrap, and refrigerate. Bring to room temperature before serving.).