Make the Garlic Aioli: Combine the pressed garlic, mayonnaise, olive oil, lemon juice, and sea salt and pepper to taste in a blender or food processor and process until smooth and creamy. Allow the aioli to sit in the refrigerator for at least an hour before serving. This can be made ahead.
Make the Patatas Bravas: Boil the fingerling potatoes in salted water until they are just tender, about 10 minutes. Drain and let dry.
Carefully slice the potatoes in half lengthwise once they are cooled.
Heat a cast iron pan with 3 tablespoons of olive oil until very hot. Work in batches (don’t overcrowd the pan) and fry up the potatoes until each side is crispy and evenly browned, about 2 minutes per side. Using tongs, transfer the potatoes to drain on paper towels, and while they are still very hot, sprinkle with sea salt, pepper, paprika, and fresh thyme.
Serve the potatoes immediately while still warm on a bed of fresh parsley, sprinkled with additional parsley and thyme. Serve with the aioli. Enjoy!