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.
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 and adjust seasoning as desired. Pour two-thirds of the dressing over the cabbage and carrot then mix well.
If the slaw seems dry, add a little more of the dressing. Let it sit in the refrigerator for about an hour to let the flavors mingle and the cabbage to soften.