In a large stockpot, heat the olive oil over medium heat, and when hot, add the onion, carrot, and celery. Sauté, stirring often, until the vegetables are very soft and starting to turn golden, about 10 to 15 minutes. Add the minced garlic and parsley to the other vegetables and cook just until fragrant, taking care not to burn the garlic, about 2 minutes
Pass the tomatoes through a passo tutto (food mill). Alternatively, crush the tomatoes in a food processor fitted with the metal chopping blade, with 3 or 4 short pulses until the tomatoes are coarsely chopped, or slightly longer for a finer puree.
Add the tomatoes to the stock pot and cook over medium heat for 1 to 2 hours, until the sauce is rich and sweet.