Step 1Heat butter in 3-quart pan over medium-high heat. When hot, add onion and cook, stirring until soft, about 6 minutes.
Step 2Add fennel and parsnips to pan along with 1 1/4 cans broth, salt and dash pepper. Bring to boil, then reduce heat and simmer, covered, until all vegetables are very soft, about 25 minutes. Let cool to room temperature.
Step 3Strain vegetables from liquid, reserving both. Puree vegetables in blender with 1/4 cup reserved greens. Add some liquid and puree until mixture is very smooth, about 1 to 2 minutes.
Step 4Return to pot. Stir in cream. (Can be made ahead up to this point and refrigerated 2 days or frozen up to 1 month. To serve, gently reheat until hot; do not boil. Add more broth or water if soup is too thick.) Taste; adjust seasoning. Serve hot garnished with reserved greens.