You will ideally want to use flat-leaf Italian parsley here, and for the nuts you can use whatever. Walnuts are traditional with parsley pesto, but pecans and pine nuts are also great.
Put all the ingredients except for the oil into a large mortar and grind to a stiff paste. Once everything is pretty well ground, drizzle in about a quarter of the olive oil and grind it into the pesto. Repeat with another quarter of the oil and then another, until you have incorporated all the oil into the pesto.
Alternately, you can put everything but the oil into the bowl of a food processor. Buzz the mixture while slowly drizzling in the oil. The more oil you use, the more like a sauce and less like a condiment your pesto will be.
When the pesto is coming together, taste for salt, and add if needed. Serve your parsley pesto right away, or put in a covered container with plastic wrap set down right on the top of the pesto to keep out air. This help keeps the color nice and bright. Use within a couple days, or freeze for up to 6 months.
Notes
Note:If you want to go an extra step, keep the parsley on the stalk and dip it in boiling salty water for 30 seconds, then shock in ice water. This sets the color and will keep your pesto green much longer.