Bring a large pot of lightly salted water to a boil. Add the pasta and cook until al dente, according to the package directions. Drain and set aside.
Meanwhile, add the olive oil to a large skillet set over medium heat. When hot, add the onion and cook, stirring occasionally, until it starts to soften, about 2 minutes. Add the garlic and cook for 30 seconds, until fragrant. Add the mushrooms and cook until they're tender and have released most of their juices, 5 to 7 minutes. Sprinkle in the flour and cook, stirring constantly, for 2 minutes. Slowly pour in the red wine, stirring the entire time. Cook, stirring frequently, until most of the wine has evaporated, 2 to 3 minutes.
Add the broth, Tamari, Worcestershire, liquid smoke, salt, smoked paprika and pepper. Increase the heat to medium high and bring the mixture almost to a boil. Turn the heat down slightly and let the mixture continue to simmer, stirring occasionally, until it thickens, 5 to 7 minutes. Turn off the heat. Stir in the Greek yogurt.
Add the drained pasta to the skillet with the sauce and toss to combine. If desired, let stand for about 5 minutes before serving (the sauce will thicken even more). Garnish with the fresh parsley and serve.