Bring a large pan of water to a boil. Cook the linguine and thyme for 5 minutes before adding the leeks and garlic. Simmer for a further 5 minutes, then add the peas. Cook for an additional 2 minutes, or until al dente.
Remove from the heat, drain, and then return to the pan. Discard the thyme sprigs. Stir through the Parmesan and cream.
Divide the pasta between 4 bowls and scatter over the parsley and a little freshly ground black pepper. Serve immediately.