Heat olive oil in a large skillet or Dutch oven over medium-high heat. Add the sliced fennel and saute until caramelized and brown (around 12 minutes).
Add broth, salt, and pepper and simmer for another 10-13 minutes until flavors meld.
Choose between a broth or a creamy soup by blending. Puree the soup in batches with a high-speed or immersion blender for a smooth texture.
Ladle into bowls, top with parsley, and serve immediately.
Notes
Store any leftovers in an airtight container in the fridge and use within several days. Reheat as needed over medium heat on the stove.