Heat the oil in a soup pot. Add the onion, carrots, and celery and sauté over low heat until the onions are golden.
Add the celery leaves and sweet potato dice. Add just enough water to cover all but about an inch of the vegetables. Bring to a simmer, then stir in the bay leaves and seasonings.
Simmer gently, covered, until the sweet potatoes and vegetables are tender, about 20 to 25 minutes. Remove the bay leaves.
With a slotted spoon, remove about half of the solid ingredients and transfer to a food processor along with about ½ cup of the cooking liquid.
Process until smoothly pureed, then stir back into the soup pot. Or, simply insert an immersion blender into the pot and process until smoothly pureed.
Add the rice milk or coconut as needed to give the soup a slightly thick consistency. Season with salt and pepper. Simmer over very low heat for another 10 to 15 minutes.
Serve at once, or let stand off the heat for an hour or two before serving, then heat through as needed. Garnish individual servings with sprigs of fresh parsley, or chopped leaves.