Set the Instant Pot to Sauté (Normal). When hot, add 1 tablespoon olive oil and 1 tablespoon unsalted butter until melted.
Pat the chicken thighs dry and season lightly with salt and pepper.
Place the chicken thighs skin-side down in the hot pot and brown for 2–3 minutes until the skin begins to turn golden; flip and brown the other side for about 2 minutes.
Turn the Instant Pot off. Place the trivet in the pot and arrange the browned chicken thighs skin-side up on the trivet without overlapping.
Pour 1 cup water or chicken stock into the pot around the trivet. Secure the lid and set to Manual/Pressure Cook (High) for 10 minutes with the valve set to sealing.
When cooking finishes, perform a quick release by carefully switching the valve to venting. Remove the chicken and trivet; discard the cooking liquid.
Set the Instant Pot to Sauté (Normal) again. Add 2 tablespoons unsalted butter and 2 tablespoons olive oil. Stir in the minced garlic and red pepper flakes if using, and cook until the garlic is fragrant and lightly golden.
Stir in 1/4 cup freshly grated Parmesan and 2 tablespoons chopped parsley until combined and the sauce is smooth.
Return the chicken thighs to the pot and toss gently to coat all sides with the garlic-Parmesan sauce. Serve immediately.
