Boil pasta in salted water until al dente, 8–10 minutes. Drain and rinse briefly to cool.
Spread pasta on a sheet pan and drizzle with 1 tbsp olive oil. Toss to prevent sticking, then let cool.
Toast pine nuts in a dry skillet over medium heat until golden, 2–3 minutes. Set aside to cool.
Whisk pesto with lemon juice and the remaining olive oil in a large mixing bowl until glossy.
Add cooled pasta, tomatoes, mozzarella, onion, olives, spinach, and basil to the bowl.
Toss until evenly coated. Season with red pepper flakes, salt, and black pepper to taste.
Fold in Parmesan and pine nuts. Chill 10–15 minutes if time allows, then serve.