Select the Sauté setting on the Instant Pot. After a minute or two, add the olive oil. Once the oil is hot, add the onions and cook for 5 minutes, stirring occasionally, until softened and lightly browned. Add the minced garlic and stir frequently for 1-2 minutes.
Add the smoked paprika, cumin, and dried chili peppers, and tomato paste. Cook, stirring frequently to prevent sticking, for 1-2 minutes, until tomato paste is darker in color.
Pour in some of the vegetable broth to deglaze the pot. Use a wooden spoon or flat-end spatula to scrape up the browned bits on the bottom of the pot.
Pour in the remaining broth, along with the water, orzo chickpeas, bay leaves, rosemary sprigs, oregano sprigs, salt, pepper, and diced tomatoes. Stir to combine.
Secure the lid and select the Pressure Cook setting at high pressure and set the cook time to 4 minutes.
After the timer is done, allow a natural pressure release for 5 minutes, then release the pressure manually. Open the pot and remove the bay leaves and any remaining whole herb sprigs. Add in the chopped sun-dried tomatoes, capers, and parsley. Stir to combine. Season to taste with salt and pepper as needed.
Transfer the soup to bowls, and if desired, garnish with additional chopped parsley and a very light sprinkling of lemon zest.