Soak the wheat berries overnight, or, as an alternative, place them in a saucepan half filled with water and boil them for 2 minutes, then remove them from the heat and let sit 1 hour. In either case, drain them, then refill the saucepan halfway with water and cook the wheat berries, partially covered, for 1 hour. When done, the wheat berries will be tender but slightly crunchy. Drain them thoroughly and let them cool.
Bring 2 cups of water, the canola oil, and salt to a boil in a mediumsize saucepan. Stir in the lentils and rice and reduce the heat to low. Cover the pan and cook for 17 minutes, or until all the liquid is absorbed. Do not stir the mixture at all while it is cooking. Carefully spoon the mixture into a large bowl and let cool to room temperature.
Meanwhile combine the dressing ingredients in a jar with a tightfitting lid and shake vigorously. Set aside.
When the rice mixture is completely cooled, use a large spoon to break up any clumps that have formed. Carefully stir in the wheat berries, carrot, red pepper, and parsley. Pour the dressing on the salad and toss. Let marinate at least 20 minutes before serving. (Note The rice greedily soaks up the dressing, so if you want to make this salad 2 to 24 hours in advance, pour on only half the dressing at first, then add the remainder about 20 minutes before serving.)