In a large pot, heat oil over medium heat. Add onion, carrots, and garlic; cook, stirring occasionally, until softened, about 5 minutes. Add lentils and thyme; cook, stirring, until lentils are coated, about 2 minutes.
Break up tomatoes using your hands or a spoon and then add along with puree. Add 4 cups water and bring to a boil. Reduce heat, cover tightly, and simmer, stirring occasionally, until lentils are tender and starting to break down, about 45 minutes. Season generously with salt and pepper. Remove from heat. If eating right away, stir in yogurt.
Meanwhile, in a large pot of boiling salted water, cook pasta until al dente according to package instructions; drain. Toss with sauce and top with grated Parmesan if desired.