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. 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 right away or let it sit in the refrigerator for about an hour to let the flavors mingle and the cabbage to soften.
