Bring a large pot of salted water to a boil. Cook the pasta until just al dente, typically 1 minute less than the package says. Reserve 1 cup (240ml) of the cooking water, then drain.
While the pasta cooks, add the roasted peppers, almonds, garlic, bread, vinegar, paprika, olive oil, and salt to a blender. Blend for about 1 minute, until thick and mostly smooth with a little texture from the almonds.
Scrape the romesco into a large skillet over medium-low heat. Warm it for 2 minutes, stirring, until it smells toasty and the surface looks glossy.
Pour in the cream and stir. Bring to a gentle simmer for 2 to 3 minutes, until the sauce turns a deeper coral and coats the back of a spoon.
Add the shredded chicken and stir for 2 minutes, until warmed through and coated.
Add the drained pasta and toss. Pour in 1/4 cup (60ml) of the reserved water and stir vigorously for 30 seconds. Add more water by the tablespoon until the sauce looks smooth, glossy, and clings to the pasta without pooling.
Remove from the heat. Stir in the Parmesan, taste, and add salt and pepper as needed.
Serve immediately, topped with parsley and extra Parmesan.