Cut the cabbages in half and then in quarters and cut out the cores. Set up the food processor with the slicing blade (according to manufacturer's instructions) and place the pieces of cabbage, one at a time, lying horizontally in the feed tube. (If they don't fit, cut them to fit lying down.) Place the feed tube pusher on top and turn on the processor. Don't push on the feed tube pusher or the slices will turn out too thick! Continue with the remaining red and green cabbage quarters. Transfer into a large bowl, discarding any very large pieces. Before you pour the dressing on the salad, save a handful of the grated vegetables to decorate for serving.
Change the slicing blade for the large shredding blade and cut the carrots so they also lie down in the feed tube. Since the carrots are hard, replace the feed tube pusher and press firmly with the food processor on. Transfer to bowl with the cabbages.
In a medium bowl, whisk together the mayonnaise, both mustards, vinegar, celery salt, kosher salt, and pepper. Pour enough mayonnaise dressing over the grated vegetables and toss to moisten well. Add crumbled blue cheese and parsley and toss together. Cover the bowl with plastic wrap and refrigerate for several hours to allow the flavors to meld. Serve cold or at room temperature.