Place cucumber in a colander over a bowl; sprinkle with salt. Toss gently to coat. Cover and chill 1 hour. Rinse with cold water; drain well. Squeeze until barely moist.
Spoon yogurt onto several layers of heavy-duty paper towels; spread to 1/2-inch thickness. Cover with additional paper towels; let stand 5 minutes. Scrape into a bowl using a rubber spatula.
Combine cucumber, yogurt, parsley, and remaining ingredients in a bowl. Serve with Mother's Souvlakia.