Wash and thinly slice the leeks into rings. Slice the onion.
Peel and slice the carrots into 1-inch slices.
Peel and cube the potato. You don’t need to be precise in your chopping, since you will puree the soup at the end.
Melt the butter in a heavy soup pot over medium heat. Add the leeks and onion. Cook, stirring until softened but not browned, about 10 minutes.
Add the vegetable broth, the carrots, potato, thyme, and bay leaf. Simmer, covered, until the vegetables are very soft, 35-40 minutes.
Remove from the heat and let cool a little. Discard the bay leaf.
Puree the soup with a stick immersion blender. Stir in the oatmilk. Season with salt and pepper.
Reheat the soup and serve sprinkled with minced parsley.