Sort the peas and remove any damaged peas, stones, or debris. Rinse thoroughly.
In a large pot, add the peas and cover with water about 2 inches above them. Bring to a boil and let them cook for 2-3 minutes. Drain carefully, rinse them and set aside.
Wash the same pot, add the peas and freshly boiled water about 4 inches above them. Bring to a boil and reduce heat to medium-high for 30 minutes. Drain carefully and set aside.
Meanwhile, in a large pan with lid, heat ¼ olive oil over medium heat. Sauté the onion until soft for 5 minutes, the garlic for another 2 minutes, the carrots for about 3 minutes and stir in the black-eyed peas for another 3-5 minutes. Stir constantly.
Pour in hot water to cover the vegetables and bring to a boil. Add the tomatoes (or passata) and the bay leaves. Season to taste. Cover with the lid, reduce the heat to low and simmer for another 20 minutes. Stir occasionally. Taste the peas. They should be tender. If not, let them simmer until you reach desired tenderness. You may add more hot water along with the cooking procedure.
When the peas are ready, turn off the heat and add the rest of the oil olive (1/4 cup) and the parsley. Stir, remove from heat and serve.
Make-ahead a good amount of black-eyed peas. Keep them in the fridge in a sealed food container for 3 days or freeze them. Use them in soups, stews, salads, etc.