Fill a large bowl with ice water. Place basil and parsley in the water and let soak for 5 minutes.
Add walnuts, olive oil, garlic and a pinch of salt to a food processor and blitz until smooth. Take the herbs from the water and shake off any excess (but don’t fully dry). Add herbs to the food processor (tearing any big basil leaves) and blitz until evenly combined (about 5 to 6 pulses). Add the cheese, lemon zest and lemon juice and blitz until just combined (about 5 more pulses). If the pesto is too thick, pulse some of the cold soaking water (a few teaspoons at a time) until it’s a smooth, creamy texture. Season to taste with additional salt.
Transfer pesto to a jar and top with a thin layer of olive oil. You can store pesto in the fridge for 1 to 2 weeks. After each time you use it, make sure there’s a thin layer of olive oil on top of the pesto. This will ensure that the top doesn’t turn brown.