Heat a large pot over medium-high. Add 1 to 2 tablespoons of olive oil along with the onion and cook, stirring, until softened, 2 to 3 minutes. Add the garlic, celery, half the parsley and ½ teaspoon salt and stir until fragrant, 1 to 2 minutes.
Add the pearl barley and stock and bring to boil. Cover, reduce heat to medium and cook, stirring occasionally. Add chopped greens after about 20 minutes, and continue cooking until the barley is swollen and tender, another 10 minutes or so.
Turn off the heat. Add the lemon juice, and season with salt and pepper. Taste and make adjust salt and lemon as needed.
Ladle the soup into bowls. Top with a dollop of yogurt and the remaining parsley and drizzle generously with olive oil.