Wash and core the cabbage and peel the carrots and beets with a vegetable peeler. Using a food processor, with the slicing disc in place, process the cabbage and transfer to a large bowl. Install the shredding disc and process the carrots and beets and transfer to the bowl. Install the chopping blade and process the parsley and transfer to the bowl. Thinly slice the red onion and add to the bowl.
Combine vinegar, oil, salt and pepper in a small bowl and whisk until salt dissolves. Toss vinaigrette with the cabbage-beet-carrot mixture. Allow the slaw to stand for 20-30 minutes before serving, tossing regularly. Add additional salt and pepper to taste and serve.
* If you do not own a food processor you can slice the cabbage into thin strips with a knife. I recommending using a hand-held or box grater to shred the beets and carrots.