Bring lentils and vegetable broth to a boil in a large pot over high heat. Reduce the heat to medium and simmer for 10 minutes.

Meanwhile, heat olive oil in a large skillet over medium-high heat. Add sausage, onion, celery, and garlic; cook and stir until sausage is crumbly and no longer pink, about 10 minutes. Stir in tomatoes, turmeric, cumin, thyme, and red pepper flakes; cook for 5 more minutes.

Stir sausage mixture into simmering lentils. Continue to simmer until lentils are tender, 20 to 30 minutes. Season to taste with salt. Garnish each serving with a dollop of yogurt and a sprinkle of chopped parsley.