You do not need a permit to visit Lahaul Spiti if you are an Indian citizen. The only permit needed in the region by an Indian citizen  is if they are crossing the Rohtang pass to enter Spiti from Manali. A vehicle permit is needed in such cases to cross the pass whether you are coming by your own vehicle, or with a rented one.
For foreign nationals, the Inner Line Permit is needed for visiting the upper parts of Lahaul-Spiti and Kinnaur valley, as these are all protected areas near international borders.