Heat the olive oil in a small saucepan over low heat. When the oil just begins to warm, add the garlic and anchovies and cook slowly, stirring, until the garlic becomes toasty brown and the anchovies dissolve, about 10 minutes. Let cool completely.

Process the butter in a food processor until smooth and creamy. Add the cooled garlic-anchovy mixture and a pinch of salt. Process until well blended. Taste and add more salt if needed. Transfer to a bowl and stir in the parsley.

Refrigerate until firm enough to shape into a log. Put an 18-inch sheet of aluminum foil on your work surface. Spoon the butter down the center of the foil into a log about 1 1/2 inches in diameter. Enclose in foil and twist the ends to make a sealed log, like a Tootsie Roll. Refrigerate for up to 1 week, or freeze for up to 6 months.