Stir together paprika, 1 teaspoon salt, and 1/4 teaspoon pepper in a bowl; sprinkle evenly over chicken. Add 1 tablespoon of the oil to a programmable pressure multicooker (such as Instant Pot). Select SAUTE setting on cooker. Select HIGH temperature setting and allow to preheat.

Add 2 chicken breasts, meaty side down, to cooker; cook until golden brown, 3 to 4 minutes. Transfer chicken from cooker to a plate. (Chicken will not be cooked through.) Repeat with 1 tablespoon oil and remaining 2 chicken breasts.

Add garlic and thyme to cooker; cook, undisturbed, 1 minute. Stir in stock, barley, and wine, scraping up any browned bits from the bottom of the cooker. Return chicken to cooker.

Press CANCEL. Cover cooker with lid, and lock in place. Turn steam release handle to SEALING position. Select MANUAL/PRESSURE COOK setting. Select HIGH pressure for 15 minutes. (It will take 4 to 5 minutes for cooker to come up to pressure before cooking begins.)

Carefully turn steam release handle to VENTING position and let steam fully escape. (This will take 2 to 3 minutes.) Remove lid from cooker; turn cooker off. Transfer chicken to a cutting board; let cool about 5 minutes. Remove and discard bones. Shred meat into large pieces.

While chicken cools, whisk together parsley, vinegar, remaining 1/2 teaspoon salt, 1/4 teaspoon pepper, and 2 tablespoons oil in a small bowl until combined.

Toss together shredded chicken and vinaigrette mixture in a large bowl. Spoon barley evenly into 6 bowls; top with chicken mixture. Serve hot.