In a large Dutch oven or soup pot, heat the oil over medium-high heat. Add the sausage, breaking up meat with back of spoon into small pieces. Cook until brown, about 7-8 minutes. Using a slotted spoon, remove the sausage to a plate lined with paper towels and reserve. (If the pot seems dry after removing the sausage, add 1-2 tablespoons of olive oil to to the bottom of the pan.)
Add the sliced leeks and fennel to the oil remaining in the pot. Sprinkle the leeks and fennel with 1/4 teaspoon of salt and sauté for about 5 minutes, stirring frequently.
Add the barley to the leeks and fennel in the pot and cook, stirring frequently until lightly toasted, about 4 minutes.
Add the broth, and stir, scraping up any bits stuck to the bottom of the pan. Add the carrots, parsnips, and cheese rind (if using), bay leaf, 1/2 teaspoon salt, and pepper. Return the sausage pieces to the soup.
Bring the mixture to a boil. Reduce the heat to simmer and cook, uncovered for about 35 minutes, or until the barley is just tender. If any foam develops on the top of the cooking soup mixture, skim it off and discard it.
If the soup seems too thick, add additional broth or water. Allow the mixture to warm up.
Add the zucchini and chard and cook an additional 10 minutes.
Remove the Parmesan rind and bay leaf. Taste for seasoning, adding more salt or pepper as needed.
Ladle into serving bowls and garnish with chives (or green onions).