Heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Add the onion and sauté until they start to soften, about 2 minutes. Add the garlic and sauté for 30 seconds. Add the mushrooms, salt and pepper and cook, stirring often, until the mushrooms start to brown, about 8 minutes. Add 4 tablespoons of the San-J Teriyaki Sauce and the rice vinegar and cook until the liquid is absorbed, about 2 minutes. Add the parsley and stir to combine.
Put the mixture into a food processor with the remaining 2 tablespoons olive oil and the remaining 1 tablespoon San-J Teriyaki Sauce. Process until the mixture is fully combined and fairly smooth. Taste and add more salt and pepper if needed.