Whisk buttermilk and 2 tablespoons oil in a small bowl; season dressing with kosher salt and pepper. Toss apple, parsnip, arugula, and dill in a large bowl. Drizzle salad with vinegar and toss to coat; season with kosher salt, pepper, and more vinegar, if desired. Spoon half of dressing onto serving plates or a large platter. Add remaining 2 tablespoons oil to salad and toss to coat. Mound salad on top of dressing. Spoon remaining dressing over and season with sea salt. Do ahead: Dressing can be made 2 days ahead. Cover and chill.