Heat a small heavy saucepan over medium heat. Add the olive oil, then add the onions and cook, stirring occasionally, for about 6 minutes, or until tender. Stir in the brown sugar and tomato paste, then stir in the pureed tomatoes and vinegar. Bring the mixture to a boil over high heat, stirring often. Reduce the heat to medium-low and simmer, stirring occasionally to prevent the ketchup from scorching, for about I hour, or until thickened to the desired consistency.

Position a rack in the middle of the oven and preheat the oven to 350°F. Line a large rimmed baking sheet with aluminum foil (for easy cleanup). Place a wire cooling rack on the foil.

Mash the bread with your hands. Add the ground beef, parsley, onions, Parmesan cheese, tomato paste, eggs, thyme, rosemary, salt, and pepper. Using your hands, mix just until all of the ingredients are thoroughly combined. Shape the meat mixture into a 9 1/2x4 1/2-inch oval loaf and place on the rack on the baking sheet. Bake for 40 minutes.

Spread half of the ketchup mixture over the meat loaf and continue baking for about 20 minutes more, or until an instant-read thermometer inserted into the center of the loaf reads 150°F. Let stand at room temperature for 10 minutes.