Rinse the split peas and barley in a fine-mesh sieve under cold water until it runs mostly clear. Dice the onion, carrots and celery, and mince the garlic.
Heat the olive oil in a 5.5-quart / 5 L Dutch oven (26 cm / 10-inch) over medium heat. Add the onion, carrots, celery and 1/2 teaspoon salt, and cook for 8 minutes, stirring, until the onion is soft and translucent.
Stir in the garlic, tomato paste, smoked paprika, thyme and black pepper. Cook for 1 minute until the paste darkens to brick red and the pot smells smoky.
Add the split peas, barley, broth, bay leaves and soy sauce. Bring to a boil over high heat, about 6 minutes, and skim off any foam. Lower the heat to a steady simmer, partially cover, and cook for 45 minutes. Stir and scrape the bottom every 10 minutes. The stew is ready when the peas have broken down into a thick base and the barley is tender with a slight chew. Add a splash of hot water if it gets too thick.
Stir in the kale and simmer for 5 minutes until wilted and tender. Remove the bay leaves, then stir in the apple cider vinegar. Taste and add salt if needed. Ladle into bowls and top with parsley.