Place the currants in a medium bowl and cover with boiling water. Let sit until soft and plump, about 5 minutes. Drain and set aside.
Meanwhile, mix the vinegar, honey, 1/4 teaspoon salt and a few grinds of pepper in a large bowl. Drizzle in the olive oil, whisking constantly, until emulsified.
Shred the broccoli florets and stems and red cabbage using the shredding disc of a food processor (see Cook's Note). Add to the bowl with the dressing and fold in the currants, almonds, onions, cilantro, parsley, 3/4 teaspoon salt and a few grinds of black pepper. Transfer to a serving bowl and garnish with more cilantro and parsley leaves, if desired.
As an alternative to shredding the broccoli, substitute about 6 cups of pre-shredded broccoli slaw.