In a large 5-quart saucepan with a tight-fitting lid (that will be used later), cook bacon over medium heat, uncovered, stirring until crispy, about 5 minutes. With a slotted spoon, transfer to a paper towel to drain. Set aside.
Add onion to saucepan; stir until golden, about 1 minute. Add mushrooms; cover. Cook until juices have evaporated, about 20 minutes.
Stir in tomato paste, thyme, and 2 cups water until combined; cover. Cook until sauce has thickened, 10 to 15 minutes. Stir in parsley and vinegar. Add bacon, reserving some for garnish, if desired.
Meanwhile, cook pasta until al dente, according to package instructions. Drain and return to pot.
To serve, toss pasta with ragout. Garnish with reserved bacon and shaved Parmesan, if desired.