Heat the oven to 450°F (232°C). Add the potatoes to a large pot and cover with cold water by 1 inch. Stir in the baking soda and 1 teaspoon salt. Bring to a boil and simmer for 7 to 9 minutes, until the edges are tender but the centers still hold.

Drain well and let the potatoes steam dry for 2 minutes. Return them to the pot, add the 1/4 cup olive oil and remaining teaspoon salt, and shake firmly until the edges look roughed and coated with a starchy paste.

Spread the potatoes on a large rimmed sheet pan with space between pieces. Roast for 20 minutes, turn, then roast for 15 to 20 minutes more until deeply golden and crisp.

Meanwhile, heat 3 tablespoons oil in a small saucepan over medium-low. Cook the garlic for 30 seconds. Whisk in the flour for 1 minute, then the sweet paprika, smoked paprika, and cayenne for 15 seconds. Gradually whisk in the warm stock.

Simmer the sauce for 4 to 6 minutes until glossy and thick enough to coat a spoon. Whisk in the sherry vinegar and salt. Taste and adjust cayenne, vinegar, or salt.

Pile the hot potatoes onto a platter. Spoon over enough salsa brava to coat some edges while leaving crisp areas exposed. Add aioli if using, scatter with parsley, and serve immediately.