Season beef cubes with salt and pepper.
Heat 1 tablespoon of olive oil in a 6 quart Dutch oven or large heavy-bottomed soup pot over medium high heat. Cook the beef in 3 batches until browned on all sides (about 3 to 4 minutes for each batch) and transfer to a plate or large bowl. Do not overcrowd the pot when cooking the beef and add more olive oil when necessary between batches. Set the plate (or bowl) of browned beef aside.
Decrease heat to medium and add onions, celery, and carrots (add more olive oil if needed). Cook for 3 minutes or until onions are translucent (stirring frequently). Add garlic and cook until fragrant (about 30 seconds), stirring constantly.
Add beef broth, tomatoes with juices, tomato paste, browned beef, Italian seasoning, and bay leaves. Stir to combine, scraping up any browned bits on the bottom of the pot.
Increase heat to medium high and bring to just a boil, then reduce heat as necessary to maintain a simmer (medium low to low). Cover and simmer for 30 minutes.
Add potatoes and stir to combine. Simmer for 15 minutes (covered).
Stir in frozen green beans, corn, and peas. Bring back to a simmer and simmer for an additional 5 to 10 minutes (uncovered) or until vegetables and beef are tender.
Season with salt and pepper to taste. Remove and discard bay leaves before serving. Serve sprinkled with parsley, if desired.