Bring a large pot of salted water to a boil. Cook the gluten free linguine according to package directions until al dente. Reserve 3/4 cup of pasta water before draining.
While the pasta cooks, pat the shrimp dry and season with salt, pepper, and red pepper flakes.
Heat 1 tablespoon of the butter and the olive oil in a large skillet over medium-high heat. Add the shrimp in a single layer and cook for about 1 1/2 minutes per side until pink and just cooked through. Transfer the shrimp to a plate.
Reduce the heat to medium. Add the remaining 3 tablespoons of butter to the skillet. Once melted, add the garlic and cook for about 1 minute until fragrant and just starting to turn golden.
Pour in the white wine and lemon juice. Let the mixture simmer for 2 to 3 minutes until the liquid reduces by about half, scraping up any browned bits from the bottom of the pan.
Add the drained linguine to the skillet and toss to coat in the sauce, adding reserved pasta water a few tablespoons at a time until the sauce is silky and clings to the pasta.
Return the shrimp to the skillet and toss gently. Finish with the lemon zest, fresh parsley, and parmesan. Serve immediately.