Preheat the oven to 325°F (160°C) and arrange the baguettes slices evenly on a baking tray.
Bake in the oven for 6-8 minutes or until lightly toasted. Remove from the oven.
In a large mixing bowl, combine ¼ cup of the mayonnaise with the tuna, shallot, lemon juice, olive oil, salt, and half of the parsley. Stir everything together and taste. Adjust for salt, pepper, lemon, and oil as appropriate.
Stuff this tuna mixture into each of the piquillo peppers, taking care not to tear the peppers. Be sure not to overstuff them!
Place each tuna stuffed pepper on a slice of toasted bread and dress with a bit more extra virgin olive oil. Optionally, top with a dollop of mayonnaise and the remaining parsley. Serve immediately.