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 ten 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 one week, or freeze for up to six months.
Add the garlic just when the oil starts to warm up. If you put the garlic in hot oil, it caramelizes too rapidly and won't flavor the oil as deeply. In the Piedmont region of Italy, eating bagna-cauda butter is a favorite ritual. The diners gather around an earthenware pot filled with hot olive oil, anchovies and garlic, and they dip raw vegetables or bread into the "hot bath", a kind of Italian fondue. The same seasonings make an incredible butter for flavoring cauliflower, broccoli or stuffed baked tomatoes. Let a nugget melt onto a grilled beef steak or fish steak, or toss a generous knob into steamed clams at the last minute. Because the dish is so simple, it's critical to use extra-virgin olive oil and the best, meatiest anchovies you can find.