In Himachal Pradesh, foreign nationals are required to get Inner Line Permits for visiting these places Khab, Samdo, Dhankar, Tabo, Gompa, Kaza, Morang, and Dubling as per the official website of Lahaul – Spiti region. You can travel to tourist places of Kinnaur Valley without any permit which are Kalpa, Sangla Valley, Chitkul and Sarahan. These places lie below the Inner Line region in Himachal, and hence, no permits are required.
However, no permit is required to visit Spiti Valley for an Indian National.
The only permit an Indian National may require can be - a vehicle permit to cross Rohtang Pass - if he/she decides to enter Spiti Valley from Manali by his/her own vehicle or self driven rented vehicle in case you are traveling from Manali. If going from Shimla, then this permit too is not required. This permit is available online. No permit is required for the reverse direction - i.e from Spiti Valley to Manali.
When you travel from Reckong Peo further to the upper region of Kinnaur and Spiti, there is a place called Jangi in Kinnaur Valley (after Kalpa / Reckong Peo) beyond which up to Kaza in Spiti Valley, a foreign national is required to get the inner line permits.