For the Slaw Mix: Combine cabbage, onion, carrot, and parsley in a large bowl, leaving plenty of room to toss (you may have to use 2 large bowls if your bowls are not large enough). Sprinkle with sugar and salt and toss to combine. Let rest 5 minutes, then transfer to a large colander and rinse thoroughly under cold running water.
Transfer rinsed mixture to a salad spinner and spin dry. Alternatively, transfer to a large rimmed baking sheet lined with a triple layer of paper towels or a clean kitchen towel and blot mixture dry with more paper towels. Return to large bowl and set aside.
For the Dressing: Combine mayonnaise, vinegar, mustard, black pepper, and sugar in a medium bowl and whisk until homogenous.
Pour dressing over cabbage mixture and toss to coat. Taste and adjust seasoning with more salt, pepper, sugar, and/or vinegar if desired.