In a large bowl, whisk together the yogurt, vinegar, and ¼ teaspoon each salt and pepper. Add the cucumbers, celery, and parsley and toss to combine.
Heat grill to medium-high. Season the salmon with ½ teaspoon salt and ¼ teaspoon pepper.
Grill the salmon until opaque throughout, 5 to 6 minutes per side. Serve with the salad.