In a large bowl whisk together the cider, orange juice, lime juice, olive oil, and sugar until the sugar dissolves. Taste and season with a pinch of salt and pepper.
Add the radishes, cucumber, celery, red onion, and cilantro or parsley if using. Gently toss to coat everything. Allow to marinate in the refrigerator for up to an hour before serving.
Arrange arugula onto serving plates or a platter, and pile the dressed salad on top. Drizzle with remaining dressing from the bowl, and serve.