Place basil, Italian parsley, and garlic into a food processor, pulse several times to combine, and process until basil is finely chopped (about 30 seconds).

Add pine nuts to basil mixture; process until finely chopped (30 seconds). Process Parmigiano-Reggiano cheese into mixture until finely ground.

Mix lemon juice into mixture with the machine running; slowly drizzle olive oil into pesto in the running machine until incorporated and pesto is thoroughly combined.