Prepare the vegetables. Finely shred the green and purple cabbage and carrots. A sharp knife, mandoline, or food processor all work well. Place them in a large mixing bowl.
Make the dressing. In a medium bowl, whisk together the mayonnaise, vinegar, sugar, Dijon mustard, celery seed, salt, and pepper until smooth and creamy. Taste and adjust seasoning as needed.
Combine the slaw. Pour the dressing over the shredded cabbage and carrots. Toss gently but thoroughly until all the vegetables are evenly coated.
Chill before serving. Cover the bowl and refrigerate for at least 30 minutes before serving. This allows the flavors to meld and the cabbage to soften slightly while staying crisp.
Serve and enjoy. Give the coleslaw a final toss just before serving. Garnish with parsley or seeds if desired.