Heat the vegetable oil in a large skillet over medium-high heat.
Add the oxtail pieces and brown them on all sides, about 5-7 minutes. Transfer browned oxtail to the slow cooker.
In the same skillet, add the chopped onion and cook until softened, about 3 minutes.
Add the minced garlic and cook for another 1 minute until fragrant.
Transfer the onion and garlic mixture to the slow cooker.
Add the sliced carrots, celery, and cubed potato to the slow cooker.
Pour in the beef broth and diced tomatoes with their juice.
Stir in the tomato paste, dried thyme, dried rosemary, bay leaves, salt, and black pepper.
Cover the slow cooker and cook on low for 8 hours, or until the oxtail is tender and the vegetables are cooked through.
Remove the bay leaves before serving.
Sprinkle chopped fresh parsley over the stew and stir gently.
Serve hot with crusty bread or over rice if desired.