Heat the olive oil in a large pot or Dutch oven over medium heat. Add the fennel and onions and simmer for 3 minutes, until they start to soften and turn golden. Add the parsnips and stir for another minute until the parsnips are coated with the oil. Add the broth and bring to a simmer. Add the beets and continue to simmer, partially covered, until the vegetables are very tender, about 35 to 45 minutes.

Allow the vegetables and broth to cool slightly. There should be about 2 cups of broth left - the liquid will have reduced in the simmering. Then transfer the contents of the pot, along with the thyme leaves, in two batches if necessary, to a food processor or blender and process to puree until smooth.

Return the soup to the pot over medium low heat and stir in the cream. Give it another moment for the cream to warm through, taste and adjust seasonings as needed. If the soup seems too thick, add more broth to reach the texture you like. Serve warm.