Cook the quails' eggs in a saucepan of boiling water for 3 minutes, then drain and cool under cold running water. Shell the eggs, then slice each in half and scoop out the yolks. Set aside.
Mash together the anchovies and oil in a bowl, stir in the lemon juice and the egg yolks and mix together until smooth, then season.
Spoon the anchovy mixture back into the halved eggs, then sprinkle over the capers and parsley and serve.