Place the zucchini noodles in a colander and toss them with 1/2 teaspoon of salt. Set aside while you make the pesto, or at least 10 minutes.
Combine the pesto ingredients except for the olive oil in a food processor, and pulse just until chopped and combined. Pulse in the olive oil and taste, adding salt and pepper if needed (this will depend on the saltiness of your Parmesan.)