Heat your Instant Pot to saute mode and add the olive oil.
Pat the chicken drumsticks dry on all sides with some kitchen paper towels.
Mix the rosemary, thyme, paprika, garlic powder, parsley, and salt and pepper in a small bowl and rub it onto the chicken drumsticks, on all sides.
Once the pot is hot, add the chicken drumsticks in one layer and let the skin brown for about 3-4 minutes on all sides.
Brown the chicken drumsticks in batches, so to not overcrowd the pot. Once one batch is done, remove them to a plate and brown the next batch on all sides.
Gently remove all the chicken drumsticks from the Instant Pot and pour in the chicken stock, scraping loose any bits off the bottom of the pot.
Place the chicken drumsticks gently on the trivet (you can layer them here) and place the trivet into the Instant Pot, and seal the lid.
Set the valve to sealing and cook on high pressure for 10 minutes.
After the cooking is finished, let the Instant Pot release naturally for 5 minutes before turning the valve to vent and release any remaining pressure.
Remove the drumsticks by lifting the trivet out of the Instant Pot and placing it on a plate for the chicken to rest for several minutes while you make the gravy. Cover the drumsticks loosely with foil as they rest.
For the gravy:
Pour the liquid left in the Instant Pot into a glass measuring cup or a bowl and skim off any unwanted fat.
Add the liquid back into the Instant Pot set to saute mode.
Whisk together the flour and water and add it to the juices in the Instant Pot, whisking well to combine.
Add the wine (if using), and continue whisking for a few minutes as the mixture thickens.
Add the paprika and salt and pepper to taste.
Turn off the Instant Pot and strain the gravy into a bowl through a fine mesh sieve to remove any large lumps or chunks.
Add the butter to the top of the gravy after it's been strained and stir the butter in gently as it melts.
Serve the drumsticks with the gravy and your favourite sides.
This recipe is great for meal prep. Simply cook as directed and store the drumsticks and gravy in airtight containers in the fridge for up to 3 days after cooking.