Bring a large pot of water to a boil for the pasta.
In a large skillet, heat the olive oil, two turns of the pan, over medium-high. Add the pancetta and cook until beginning to crisp, 2 to 3 minutes; transfer to paper towels. Add the butter to the skillet. When it foams, add the onion. Cook, stirring often, until the onion softens, about 5 minutes. Add the peas and garlic and cook until the peas are heated through, 2 to 3 minutes. Stir in the tomato paste, then add the stock and simmer until thickened slightly, a couple minutes more; add the pancetta.
Meanwhile, salt the boiling water, add the pasta and cook to al dente.
Add 1/2 cup of the pasta cooking water to the peas; drain the pasta.
Toss the pasta, cheese and parsley with the peas. Serve in shallow bowls.