Preheat the oven to 350 degrees (176 degrees C). Lightly oil a small baking sheet or baking dish with a little vegetable oil or cooking spray. Put the panko in a medium bowl, add the soy milk (or water), and stir to combine. Set aside.
Wash and dry the mushrooms. Pop the stems out of the caps, and place the caps, gill side up, on the baking sheet. Finely chop the mushroom stems, and put in a small pan with the lemon juice and 2 tablespoons of the olive oil. Pour the remaining 2 tablespoons of olive oil into the mushroom caps, evenly dividing it among them.
Put the pan with the mushroom stems on the stove, and turn the heat to medium-low. Put the baking sheet in the oven. Let them both cook for 10 minutes.
Remove the mushroom caps from the oven and set aside. (Don't turn off the oven.) Add the mushroom stems and the parsley to the panko, and stir to combine. Season to taste with salt and pepper. Divide the mixture among the mushroom caps. Put the now-stuffed mushroom caps back in the oven for 20 minutes.
Take the mushrooms out of the oven, heat the broiler, and run the mushrooms under the broiler for a minute or two until the tops are browned. Transfer to a plate and serve.