In large saucepan, heat oil over medium heat. Add onion and carrot to pan; heat 5 minutes, stirring occasionally. Add zucchini, celery, bell pepper, and garlic to pan; heat 3 minutes. Add tomato paste, Italian seasoning, salt, black pepper, and red pepper flakes and heat 1 minute. Stir in tomatoes.

Add broth, lentils, and farro to pan. Bring to a boil; reduce heat to medium-low and simmer, covered, until lentils and farro are tender but not mushy, about 30 minutes. Add additional broth or water if soup becomes too thick. Stir in spinach and red wine vinegar; heat until spinach has wilted.

Divide soup among serving bowls and garnish with chopped parsley.