In a large pot, preferably of terracotta, sauté the garlic cloves, along with the red pepper and the stems of the parsley sprigs, in the olive oil over moderate heat for just a few moments, until the garlic begins to give off its aroma.
Now add the octopus, head down, and cover. Let the octopus simmer over gentle heat for about 15 minutes.
Uncover the pot. The octopus should have given off a fair amount of liquid. Pour the tomatoes, olives and capers and re-cover tightly. (You can use some parchment paper or aluminum foil to create a better seal if you like.) Simmer the octopus gently until tender, about 30-45 minutes.
Mix in the minced parsley leaves, give the pot a turn, and let it sit for five minutes.
Serve your polpo alla luciana warm, with a nice chunk of crusty bread to soak up those delicious juices.