In a small saucepan, cover the lentils with 2 1/2 cups of water and bring to a boil. Cover and cook over low heat until the lentils are tender, about 30 minutes.
Meanwhile, in a medium saucepan, heat 2 tablespoons of the olive oil. Add the shiitake and season with salt. Cover and cook over moderate heat, stirring, until the mushrooms are tender and starting to brown, 5 minutes. Add the remaining 1 tablespoon of olive oil along with the garlic, cumin, coriander, pepper and turmeric and cook, stirring, until fragrant, 1 minute. Add the greens and cook, stirring, until wilted, 2 minutes.
Add the lentils and their cooking liquid to the mushrooms and simmer for 3 minutes. Add up to 1/4 cup of water if the lentils are too dry. Season with salt. Ladle the lentils into bowls, garnish with the parsley and serve.