In the bowl of a food processor, add arugula (in batches), parmesan, walnuts, salt, and black pepper. Turn the food processor on high and drizzle in olive oil through the feed tube until everything is combined. Remove blade from food processor and using a spatula, scrape out the pesto and store it in an airtight container or what I like to do is store it in mason jars.
In a large stock pot, bring water to a boil then add pasta and cook according to the instructions on the back of the box. Drain pasta well, reserving 1 cup of pasta water. In a large bowl, add the pasta and 4 tbsp. of the arugula walnut pesto to the pasta. Toss gently until all is combined. If mixture looks too thick, pour a little bit of pasta water on top, 1 tbsp. at a time. You probably will not use all of the pasta water. Toss the sundried tomatoes in with the pasta and add more pesto, if needed.
Sprinkle parmesan cheese on top prior to serving.
Keep pesto in fridge in an airtight container for up to two months and keep pasta in fridge in an airtight container for up to 5 days.
Notes
This makes a lot of arugula pesto but I like making a lot at one time because I store them in mason jars and keep the in the fridge for quick and easy meals on the weekdays after work. It keeps for a good couple months so it's nice to have it on hand!