Spoon yogurt into a cheesecloth-lined colander; set aside until most of water has drained, at least 30 minutes.
Place grated cucumber and grated carrot in a cheesecloth-lined colander; sprinkle with salt. Place a heavy object, such as a bowl, atop cucumber and carrot to drain excess water, 15 to 20 minutes.
Mix yogurt, cucumber, carrot, onion, red bell pepper, celery, parsley, lemon juice, mint, and cumin in a bowl; season with salt and black pepper. Refrigerate for flavors to blend, at least 1 hour.
