Warm the olive oil in a large skillet over medium-high heat, then add the diced onion. Let it cook until soft and lightly golden, about 8 minutes, stirring now and then. Stir in the carrots, celery, and spices, and cook for another 2 minutes before adding the garlic for just 30 seconds.
Scoop the sautéed vegetable mixture into your slow cooker. Add the lentils, both cans of tomatoes, the vegetable broth, and the Parmesan rind if you're using one. Give everything a good stir so the lentils are fully submerged in the liquid.
Cover the slow cooker and cook on low for 6 to 8 hours or on high for 3 to 4 hours. You'll know it's ready when the lentils are tender but still hold their shape rather than turning to mush.
Remove the Parmesan rind, then stir in the red wine vinegar and sugar to balance the flavors. Taste and adjust the seasoning, thinning with a splash of broth if the soup feels too thick, then ladle it into bowls.