Place arugula, walnuts, and garlic in the bowl of a food processor and pulse until finely chopped, about five 1-second pulses, scraping down sides of bowl as necessary.
With the food processor running, pour oil through feed tube in a slow, steady stream. Keep food processor running until smooth, stopping to scrape down sides of bowl as necessary.
Add the Parmesan and lemon juice and pulse to combine. Season with salt and pepper to taste. Store in an airtight container in the fridge for up to a week, or in a resealable plastic bag in the freezer for up to six months.