Heat oil in a large pot over medium heat. Add the onion and carrots and cook, stirring occasionally, until vegetables are tender and beginning to brown, about 5 minutes. Add garlic, thyme, smoked paprika and rosemary and cook for an additional minute.
Add split peas, barley, broth, water, baking soda, bay leaf and 1 teaspoon coarse salt to the pot. Bring to a boil and then reduce to a simmer. Continue to cook until peas are very soft and the barley is pleasantly tender and chewy, about 35-45 minutes.
Add soy sauce, lemon juice and parsley and season the soup to taste with salt and pepper. Serve, topped with a drizzle of olive oil and a sprinkle of smoked paprika.