Preheat an oven to 425°F. Spread the pistachios on a baking sheet and toast in the oven until fragrant and just turning brown. Cool, roughly chop, and transfer to a small bowl. Combine with the lemon zest, garlic, and parsley.
Meanwhile, toss the broccoli with 2 tablespoons of olive oil and salt and pepper and spread out on a large baking sheet in a single layer. Roast for 20-25 minutes, until lightly brown and beginning to crisp. Remove from the oven and sprinkle with lemon juice while still very hot.
While the broccoli is cooking, bring a large pot of salted water to boil. Cook the pasta until al dente and drain, reserving 1 cup of pasta cooking water.
Return the pasta to the pot and add the broccoli, most of the pistachio gremolata, and the remaining olive oil. Toss well to combine, adding pasta water as needed to create a viscous sauce. Season to taste with salt and pepper and divide among bowls. Top with any remaining gremolata.