Pat the beef cubes dry with paper towels and toss them in the 1/4 cup flour until lightly coated; shake off excess flour.
Heat 1 tablespoon oil in a large skillet over medium-high heat. In two batches, sear the floured beef until browned on all sides (about 2–3 minutes per side); transfer browned pieces to the slow cooker.
Reduce the skillet heat to medium, add the remaining tablespoon of oil and sauté the chopped onion until translucent, about 4 minutes. Add the garlic and cook 30 seconds more to release aroma, scraping up any browned bits.
Pour the sautéed onion and garlic into the slow cooker over the beef. Stir in the tomato paste and Worcestershire sauce so they coat the meat and aromatics.
Add 2 cups of the beef broth, 1 teaspoon salt, 1/4 teaspoon pepper, and the dried thyme to the slow cooker. Reserve the remaining 1 cup broth for later. Cover and cook on LOW for 4 hours (or HIGH for 2–3 hours).
After the initial cook time, add the carrots and potatoes to the slow cooker along with the reserved cup of broth. Taste and add the remaining 1 teaspoon salt and remaining pepper if needed. Continue cooking on LOW for 3–4 more hours (or on HIGH for 1½–2 hours), until beef is very tender and vegetables are cooked through.
If you prefer a thicker gravy, mix 2 tablespoons of the reserved cold water with 1 tablespoon flour (or 1 tablespoon cornstarch) to make a smooth slurry. Stir the slurry into the stew, switch the slow cooker to HIGH, and cook uncovered 15–20 minutes until the sauce has thickened.
Turn off the slow cooker and let the stew rest for 10 minutes before serving to allow flavors to meld and the sauce to settle.
Ladle the stew into bowls, sprinkle with chopped parsley if using, and serve warm with crusty bread or over buttered egg noodles.