Process feta and buttermilk in a food processor until smooth. Add cilantro, parsley, tarragon, anchovies (if using), and lemon zest. Continue to process, adding more buttermilk if needed, until mixture is smooth and pale green with small flecks of herbs still visible. Add lemon juice, season with salt and pepper, and process just to combine.
Transfer dip to a medium bowl. Drizzle with oil and top with a few sprigs of parsley. Serve on a platter with crudité.
Do Ahead: Dip can be made 1 day ahead. Cover and chill.