Begin by peeling and dicing the potatoes, carrots, parsnip, and celery root into small, even pieces. Chop the onion and mince the garlic.
In a large pot, heat the olive oil over medium heat. Add the chopped onion and garlic, sautéing until fragrant and translucent, about 3-4 minutes.
Toss in the diced potatoes, carrots, celery root, celery stalks, and parsnip. Sauté for another 5-7 minutes, stirring occasionally until the vegetables begin to soften.
Add the veggie broth to the pot and bring it to a simmer.
In a mixing bowl, combine the all-purpose flour, corn starch, baking powder, dried parsley, and salt. Mix well. Then, add enough water to form a soft dough.
Using your hands, form small balls of the dough, about the size of a tablespoon. Set them aside on a plate.
Once the broth is simmering, add the cannellini beans and green peas. Stir well to combine.
Gently drop the dumplings into the simmering soup. Cover the pot and let them cook for about 10-12 minutes.
Once the dumplings are cooked, stir in the fresh parsley and saffron (if using). Season with salt and pepper to taste.
Ladle the Vegan Dumpling Soup into bowls, garnishing with additional fresh parsley if desired. Serve hot and enjoy!