Combine parsley, onion, garlic, and red pepper paste in the bowl of a food processor; process until finely blended. Transfer to a large bowl. Add turkey, basil, thyme, cumin, cinnamon, allspice, salt, and pepper; mix until thoroughly combined. Form mixture into uniformly-sized meatballs.
Heat oil in a large saucepan over medium-low heat. Add meatballs, working in batches if necessary, and saute until browned on all sides, 2 to 3 minutes per side. Remove to a plate.
Add 4 cups water, spaghetti sauce, diced tomatoes, and tomato paste to the saucepan; mix to combine. Add more water if sauce seems too thick. Drop meatballs into sauce and cook over medium-low heat until flavors have melded, about 1 hour.