Boil the pasta in well-salted water until al dente, 9–10 minutes; drain and rinse briefly to cool.
Toast the pine nuts in a dry skillet over medium heat until golden and fragrant, about 3 minutes. Cool.
Whisk pesto with olive oil, lemon juice, salt, and pepper to loosen into a pourable dressing.
Combine cooled pasta, tomatoes, mozzarella, red onion, and arugula in a large bowl.
Pour in the pesto dressing and toss until evenly coated. Fold in Parmesan and toasted pine nuts.
Taste and adjust seasoning. Chill 15–30 minutes for flavors to meld, if time allows.