Add the lime juice, olive oil, apple cider vinegar, grated ginger, grated garlic, honey, Dijon, salt, and pepper to a 250 ml / 1-cup jar. Seal and shake for 20 seconds until the dressing looks cloudy and fully combined.
Quarter the cabbage, cut out the white core, and slice the wedges 2 to 3 mm thick with a sharp chef's knife or mandoline. Add the cabbage to a 4-quart / 3.8 L mixing bowl, pour in the dressing, and massage with your hands for 1 to 2 minutes, until the cabbage turns deep magenta, looks glossy, and shrinks by about a third. If it still feels stiff, keep going.
Grate the carrots and cut the apple into thin matchsticks. Add both to the bowl with the cranberries, scallions, and parsley, then toss until everything is coated.
Let the slaw rest for 15 minutes at room temperature so the cranberries plump up and the cabbage relaxes. Taste, add a pinch of salt or a squeeze of lime if needed, and fold in the pumpkin seeds right before serving.