Melt 2 tablespoons of the butter with the olive oil in a Dutch oven over medium-high heat. Add the onion and pancetta and cook until the onion is tender and the pancetta begins to brown, about 5 minutes.
Reduce to medium heat and add the carrot and celery. Cook, stirring frequently until the vegetables are tender.
Add the tomatoes and the juices from the can. Cook, stirring frequently, until the liquid is evaporated and the mixture is thickened, about 20 minutes.
Add the lentils, broth, and water. Season with salt and pepper to taste. Bring to a boil. Reduce the heat to low and cook, uncovered, maintaining a gentle simmer until the lentils are tender and the soup is to desired thickness, about 45 minutes.
Stir in ¼ cup of the Parmesan cheese and the remaining butter. Adjust the salt and pepper as needed.
To serve, ladle the soup into bowls and sprinkle with parsley and more Parmesan cheese.