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Quarter the cabbage through the core, and then cut out the core. Cut each quarter crosswise in half and finely shred. Place the shredded cabbage in a very large bowl (you will have 6 to 8 cups).
Add the shredded carrot and parsley to the cabbage and toss to mix.
In a separate bowl, stir the mayonnaise, vinegar, mustard, celery seeds, salt, and pepper together. Taste for acidity and seasoning, then adjust as desired. If the dressing tastes too tart and you prefer a sweeter coleslaw, stir in the optional sugar.
Pour two-thirds of the dressing over the cabbage and carrot, then mix well. (Clean hands are the quickest tool).
If the coleslaw seems dry, add a little more of the dressing. Eat immediately or let it sit in the refrigerator for about an hour to let the flavors mingle and the cabbage softens.