In a Dutch oven over medium high heat, cook onions, carrots and celery in 1 tbsp water for 8 minutes, stirring frequently and adding an additional 1-2 tbsp of water as needed to prevent sticking. Stir in mushrooms and garlic and cook for another 5 minutes. Stir frequently and add water as needed.
Stir in potatoes, tomato paste, Italian seasoning, paprika, rosemary, and broth. Bring to a boil and reduce heat to medium low. Cook, covered, for 15 minutes. Stir occasionally.
Stir in peas and cook for another 5 minutes, or until potatoes are tender.
Transfer 2 cups of the mixture to a blender. Blend until smooth and return mixture to Dutch oven. Stir to combine.
Remove pot from the heat and stir in parsley, salt, pepper and liquid smoke if using.
Ladle into bowls and serve.