Heat the vegetable oil in a large heavy-bottomed pot or Dutch oven over medium-high heat.
Add the skillet beef stew meat in batches, making sure not to overcrowd the pot, and brown on all sides, about 5 minutes per batch. Remove the browned meat and set aside.
In the same pot, add the chopped onion and cook until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Return the browned beef to the pot and stir to combine with the onions and garlic.
Add the sliced carrots, celery, and potatoes to the pot.
Pour in the beef broth and water, then stir in the tomato paste, dried thyme, dried rosemary, and bay leaves.
Season with salt and black pepper to taste.
Bring the stew to a boil, then reduce the heat to low and cover the pot.
Simmer gently for 1 hour and 30 minutes, or until the beef and vegetables are tender.
In a small bowl, whisk together the all-purpose flour and cold water until smooth to create a slurry.
Stir the slurry into the stew and cook uncovered for an additional 10 minutes to thicken the broth.
Remove the bay leaves and adjust seasoning with additional salt and pepper if needed.
Serve the beef stew hot, garnished with chopped fresh parsley.