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Put the black-eyed peas in a large pot and add enough water to cover them by 2 to 3 inches. Cover and bring to a boil, then adjust the heat so the water simmers gently. Cook, stirring occasionally, until the peas are mostly tender, 30 to 40 minutes; season with salt. (You can do this step a day or two ahead of time: Cool the peas, transfer them to an airtight container with their cooking liquid, and refrigerate until you’re ready to make the soup.)
Put ⅓ cup of the olive oil in a large, deep pot over medium-high heat. When it’s hot, add about half the onions, carrots, and celery and cook, stirring occasionally, until they soften, about 10 minutes. Add the remaining ⅓ cup oil and the remaining onions, carrots, and celery; season liberally with salt and pepper. Cook, stirring occasionally, for another 10 minutes.
Add the garlic, the cabbage, and most of the parsley; cook, stirring occasionally, until the cabbage is softened, 5 to 7 minutes.
Add the black-eyed peas (with their liquid) and the chopped tomatoes to the pot, along with enough water to make the mixture stewlike but not watery. Cover, bring to a boil, reduce the heat so the mixture simmers steadily, and cook until the cabbage is very tender, 20 to 30 minutes. Taste and adjust the seasoning, and serve hot or warm, garnished with the remaining parsley.