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Place the peaches in a heatproof, nonreactive bowl or mason jar. Pour the hot pickling liquid over and let cool to room temperature. Refrigerate overnight or up to 48 hours before using.
For the pasta: Cook the pasta in a large pot of boiling salted water, stirring occasionally, until al dente, according to package directions. Drain, reserving the pasta cooking liquid in the pot, and drizzle with the olive oil. Transfer to a bowl, add the asparagus and radishes and toss to combine. Add the herbs, lemon zest and pepper flakes and toss to combine again. Let cool.
Stir in the arugula, 1/4 cup pickled peaches and the Parmesan. Toss to combine, then add salt and pepper to taste and more olive oil, if needed.