Preheat your oven to 400°F and position a rack in the center.
Mix the mayonnaise with 1 tablespoon of smoked paprika in a large bowl until evenly combined. Scoop out 1/4 cup of this mixture and set it aside in a small bowl — this becomes your finishing sauce. Season the chicken pieces generously with salt and pepper, then add them to the large bowl with the remaining paprika mayo. Stir until every piece is well coated.
On a large rimmed baking sheet, toss the sliced potatoes with 2 tablespoons olive oil, the remaining teaspoon of paprika, and a good pinch of salt. Spread them in a single layer, placing the thicker slices toward the edges where they'll get more heat. Nestle the coated chicken pieces on top, spacing them evenly so they don't steam each other.
Roast until the chicken registers 165°F on an instant-read thermometer and the potato edges are golden and crispy, 20-25 minutes. The chicken should be nicely browned on top.
⏱ 25 min
While everything roasts, finish your paprika mayo by whisking the remaining tablespoon of olive oil and juice from half the lime into the reserved mayonnaise. Taste and adjust with salt and more lime juice as needed — you want it bright and tangy to cut through the richness. Keep it refrigerated until serving time.
Sprinkle the hot chicken and potatoes with chopped parsley and serve immediately with the paprika mayonnaise on the side for drizzling.