Working in batches, pulse the vegetables in a food processor until finely chopped. Transfer to a large bowl and toss well. Stir in the sugar.
In a small saucepan, combine the vinegar, oil, dry mustard, celery seeds and salt and bring to a boil. Pour the dressing over the slaw and toss well. Refrigerate overnight. Just before serving, drain the slaw and stir in the parsley.