Place the white cabbage, red cabbage, greens, red onions, green onions, and parsley in a large salad bowl. In a small bowl, combine the mayonnaise, mustard, salt, black pepper, cayenne, and sugar. Mix well. Add the mixture to the greens and toss to mix thoroughly. Cover and refrigerate for at least 1 hour. Serve chilled. The slaw can be made three hours ahead.
Recipe adapted from Louisiana Real and Rustic Cookbook, by Emeril Lagasse with Marcelle Bienvenu, published by William Morrow, 1997