Place all of the yolks in a medium bowl. Add tuna, parsley, olives, tomato, mustard, lemon juice, vinegar, and olive oil. Season to taste with salt and pepper, and using a rubber spatula, gently fold until combined.
Filling and egg white halves can be stored covered in the refrigerator up to overnight before filling and serving (lay egg white halves cut-side-down on a large plate and cover with plastic wrap).
To serve, spoon filling into egg white halves. Garnish with more parsley and drizzle with olive oil. Serve immediately.
