Brown the beef: Pat the beef cubes dry and season them all over with the salt and pepper. Heat 1 tablespoon of the oil in a large Dutch oven over medium-high heat and brown the beef in 2 batches, about 5 minutes each, turning so most sides pick up color. Don't crowd the pot. Move the browned beef to a plate.
Cook the vegetables: Add the remaining 1 tablespoon of oil, then the onion, carrots, and celery. Cook 5 minutes, scraping the browned bits off the bottom as the vegetables release their moisture.
Add the garlic and tomato paste: Stir in the garlic and tomato paste and cook 1 minute, until the paste darkens and coats the vegetables.
Build the broth: Pour in the beef broth and scrape the bottom of the pot clean. Add the diced tomatoes with their juice, the bay leaves, thyme, and Worcestershire, then return the beef and any juices from the plate. Bring it to a boil, about 4 minutes.
Simmer the beef: Turn the heat to low, cover, and simmer 1 hour 15 minutes. The beef clears the USDA's 145°F mark for beef within a few minutes of hitting the broth, but it doesn't go tender until the connective tissue breaks down closer to 200°F. Test at 1 hour 15: a cube should come apart when you press it with a fork. If it springs back, keep simmering and check every 15 minutes.
Add the potatoes, then the green beans: Stir in the potatoes and simmer uncovered 20 minutes, until a knife slides in easily. Add the green beans and simmer 10 minutes more, until they're tender and still green.
Finish: Stir in the corn and peas and cook 5 minutes, just until heated through. Fish out the bay leaves, stir in the parsley, and taste for salt before serving.
Notes
Simmer the beef covered for a full 1 hour 15 minutes before the potatoes go in. Stew meat passes the USDA's 145°F mark for beef within minutes of hitting the broth, but it doesn't turn tender until the connective tissue breaks down closer to 200°F, so judge it by a cube falling apart under fork pressure rather than by the clock alone. Keep the pot at a lazy simmer; a rolling boil toughens the meat. Reheat leftovers to 165°F.