Preheat the oven to 425°F (220°C).
Place the halved baby potatoes in a large bowl. Add 2 tablespoons of olive oil, minced garlic, dried thyme, dried rosemary, paprika, salt, and black pepper. Toss well to coat all potatoes evenly.
Spread the potatoes on a baking sheet in a single layer. Roast in the preheated oven for 20 minutes.
While the potatoes roast, pat the salmon fillets dry with paper towels. Season both sides with salt and black pepper.
After the potatoes have roasted for 20 minutes, remove the baking sheet from the oven. Push the potatoes to one side of the sheet.
Drizzle the remaining 1 tablespoon of olive oil over the salmon fillets and place them skin-side down on the empty side of the baking sheet. Arrange lemon slices on top of the salmon.
Return the baking sheet to the oven and roast for an additional 10 minutes, or until the salmon is cooked through and flakes easily with a fork, and the potatoes are golden and tender.
Remove from the oven and sprinkle fresh chopped parsley over the salmon and potatoes.
Serve immediately, garnished with extra lemon slices if desired.