In a medium bowl, fold together tuna, 1/4 cup allioli, shallots, lemon juice, capers, 2 tablespoons minced parsley, and olive oil.
Season with salt and pepper to taste, and add more lemon juice, olive oil, or allioli if needed.
Fill the piquillo peppers with the tuna mixture.
Arrange the baguette slices on a serving board. Put a filled pepper on each slice. Add a dab of allioli on top.
Sprinkle with the rest of the parsley, drizzle with olive oil, and add some coarse sea salt.
Insert a toothpick in each one to hold it to the bread and it becomes a Spanish Pincho..