Remove and discard tough outer leaves from the green cabbage. Cut the head in half, cutting through the core. Use the tip of the knife to remove the core from both halves, then cut each half into quarters. Slice each quarter thinly into short strips. Repeat with the red cabbage and place both in a large bowl.
Whisk together the mayonnaise, vinegar, sugar and celery seeds in a small bowl, then season with salt and pepper. Add more sugar as desired. Toss the cabbage with half the dressing, then lay in a shallow dish (a 9-by-13-inch baking dish works well). Cover and refrigerate for 1 to 2 hours.
Once cabbage has marinated in the dressing, taste and add reserved dressing as desired. Add the carrots and parsley and toss to combine.