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In an Instant Pot or other pressure cooker, heat the tablespoon of oil over Sauté mode until the oil is shimmering. (Use medium heat in a stovetop pressure cooker). Sprinkle the drumsticks with 1 teaspoon salt, the paprika, and the garlic powder. Brown the drumsticks in 2 batches: Add half the drumsticks in a single layer, skin side down, and cook until browned on that side, about 3 minutes. Remove the browned drumsticks to a bowl, add the rest of the drumsticks, and brown on one side, about 3 more minutes. Move to the drumstick bowl.
Add the butter to the pot and let it melt, scraping any browned bits of chicken loose from the pot with a flat edged wooden spoon. Once the butter is melted, add the potatoes, and sprinkle with ½ teaspoon fine sea salt and ½ teaspoon dried thyme. Toss the potatoes to coat with butter, then cook until the potatoes are browned on the bottom, about 3 minutes.
Pour in the 2 cups of chicken broth, and sprinkle with ½ teaspoon of salt if using homemade broth. Stack the browned drumsticks on top of the potatoes and pour in any chicken juices left in the bowl.
Lock the lid. Cook on high pressure for 15 minutes in an Instant Pot or other electric PC, or for 13 minutes in a stovetop PC. (Use “Manual” or “Pressure Cook” mode in an Instant Pot.) Let the pressure come down naturally, about 15 more minutes. (If you get impatient you can quick release the remaining pressure after 15 minutes of natural release.)
Unlock the pressure cooker (carefully – tilt the lid away from you to avoid any hot steam.) Move the drumsticks and potatoes to a platter with tongs and/or a slotted spoon. Ladle a scoop or two of the pot sauce over the drumsticks and potatoes, and sprinkle with minced parsley. Pass the rest of the pot sauce at the table. Enjoy!