Toss the octopus pieces with olive oil, minced garlic, 1 teaspoon smoked paprika, salt, and black pepper in a bowl until evenly coated.
Thread the seasoned octopus onto the soaked skewers, leaving a small gap between pieces for even cooking.
Heat a grill or grill pan over medium-high heat. Place the skewers on the hot surface and cook for 3 to 4 minutes per side until the edges char and the octopus firms up.
While the skewers cook, stir together the mayonnaise, lemon juice, and remaining 1 teaspoon smoked paprika in a small bowl to make the aioli.
Remove the skewers from the heat and let them rest for 1 minute. Serve with the smoked paprika aioli on the side, along with lemon wedges and a sprinkle of chopped parsley.