Season beef with salt and pepper, then toss with flour. Heat oil in a large Dutch oven over medium-high heat and brown beef in batches until golden on all sides, about 8 minutes total. Transfer to a plate.
Reduce heat to medium and add onion and celery to the pot. Sauté for 5 minutes until softened, then add garlic and cook for 1 minute more.
Return beef to the pot and add beef broth, water, Worcestershire sauce, bay leaves, rosemary, and thyme. Bring to a boil, then reduce heat to low.
Cover and simmer gently for 1 hour, stirring occasionally to prevent sticking.
Add parsnips and carrots to the pot, stir well, cover, and continue cooking for 50 minutes until vegetables are tender and beef is fork-tender.
Remove bay leaves and taste for seasoning, adding more salt and pepper as needed. The parsnips will have naturally thickened the broth.
Let rest for 10 minutes before serving, then garnish with fresh parsley.