Sort and wash black-eyed peas; place in a small saucepan. Cover with water to 2 inches above peas; bring to a boil. Cook 2 minutes. Remove from heat; cover and let stand for 1 hour. Drain peas, and place in a saucepan. Cover with water to 2 inches above peas. Bring to a boil; cover, reduce heat, and simmer 30 minutes or until tender. Drain.
Combine vinegar and next 5 ingredients (vinegar through salt) in a small bowl; stir well with a whisk.
Steam the beets, covered, for 25 minutes or until done. Arrange the beets on a platter, and top with peas. Sprinkle with cheese, pecans, and parsley. Drizzle the vinaigrette over the salad.
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