Preheat oven to 450°F. Remove sardines from tin, reserving sardine oil; place sardines in an even layer on a parchment paper–lined small rimmed baking sheet. Using a fork, mash sardines until finely mashed and spread out into an even layer, leaving some spaces in between. Bake until crispy, 12 to 14 minutes. Let stand for 1 minute, as sardines may pop; transfer to a paper towel-lined plate and set aside until ready to use.
Meanwhile, bring a large pot of lightly salted water to boil over high. Add spaghetti, reduce heat to medium-high; cook according to package directions for al dente, 9 to 11 minutes. Drain, reserving 1 cup pasta water.
While pasta is cooking, add enough olive oil to reserved sardine oil to measure ⅓ cup; heat in a large skillet over medium-low. Add garlic; cook, stirring often, until light golden brown, about 7 minutes. Add crushed red pepper; cook, stirring constantly, until fragrant, about 30 seconds. Add drained spaghetti to skillet over medium-low. Add ¾ cup pasta water; toss until spaghetti is thoroughly coated, adding remaining ¼ cup pasta water, 1 tablespoon at a time, as needed.
Remove from heat, add parsley and Parmigiano; toss well until evenly distributed. Garnish with crispy sardines, lemon zest, additional red pepper, parsley, and Parmigiano.
