Heat 1 tablespoon of oil in a Dutch oven over medium heat until shimmering. Add mushrooms and salt. Cover and cook for about 5 minutes until mushrooms release their liquid. Uncover and cook until mushrooms are dry and browned, about 5–10 minutes.
Stir in the remaining 1 tablespoon of oil, chard stems, onions, celery, carrot, and red bell pepper. Cook until vegetables are well browned, about 7–10 minutes. Stir in garlic, tomato paste, and thyme; cook until fragrant (about 30 seconds). Stir in flour and cook for 1 minute. Stir in wine, scraping up browned bits, and cook until nearly evaporated (about 1 minute).
Stir in broth, water, potatoes, parsnips, celery root, and bay leaves. Bring to a simmer. Reduce heat to medium-low, partially cover, and cook until vegetables are tender and the stew is thickened, about 1 hour.
Stir in zucchini and chard leaves. Simmer uncovered until just tender, about 5–10 minutes. Remove from heat and discard bay leaves. Adjust consistency with extra broth if needed. Stir in parsley and lemon juice. Season with salt and pepper to taste. Serve.