Heat 1 tablespoon oil in a large pot over medium-high heat. Add chopped squash, chopped onion, sliced carrots and sliced garlic; cook, stirring occasionally, until the vegetables begin to soften, about 5 minutes.
Add rinsed lentils, 1 teaspoon each coriander, cumin and salt, ½ teaspoon each pepper and cinnamon, ¼ teaspoon allspice and 2 bay leaves. Cook, stirring constantly, until fragrant, about 30 seconds.
Stir in 8 cups broth and 1 can tomatoes with their juice. Cover and bring to a lively simmer over medium heat. Cook, covered and undisturbed, until the lentils and squash are tender, about 20 minutes.
Remove from heat. Discard the bay leaves. Stir in ¼ cup yogurt.
Divide the soup among 6 bowls. Top with ¼ cup parsley. Serve with additional yogurt and lemon wedges, if desired.
