Bring a large pot of salted water to a boil; add the wild rice and cook until just tender, about 40 minutes; drain. Set aside and cover to keep warm.
Meanwhile, heat the olive oil in a deep skillet over medium heat. Add the sausage and cook, breaking up the meat with a wooden spoon, until browned, about 5 minutes. Pour off all but about 3 tablespoons of the drippings. Add the leeks and thyme and cook, stirring occasionally, until the leeks are just wilted, about 5 minutes. Stir in the basmati rice and cook until lightly browned, 1 to 2 minutes. Add 1 1/3 cups water and 1/4 teaspoon salt and bring to a boil. Reduce the heat to low; cover and simmer until the rice is tender, about 15 minutes. Remove from the heat and set aside, covered, 5 minutes.
Fluff the basmati rice with a fork, then stir in the prepared wild rice and the parsley. Season with salt and pepper.