Both and neither. Taekwondo began development while the Korean peninsula was under the control of the Japanese before World War II, and continued to be developed during and after the Korean War, when North and South Korea split.
Furthermore, the martial artists involved in creating Taekwondo were from different parts of the Korean peninsula. Choi Hong-hi, who is the founder of Changheon-yu Taekwondo, was born in what is now North Korea, but during and for a long time after the Korean War, was allied with government of South Korea. Son Deok-seong – a pivotal figure in the development of the Cheongdo-kwan style, was born in Seoul, which is now in South Korea.
Thus, Taekwondo is from both North Korea and South Korea, in that some of the people who created Taekwondo were from a part of the peninsula that is now in North Korea, and some from a part that is now in South Korea. It is also from neither North and South Korea, in that when Taekwondo began being developed, the separate states of North and South Korea did not exist.
Today, Taekwondo is practised in both North and South Korea, but the different nations prefer different styles. North Korea favours Choi Hong-hi’s style – Changheon-yu Taekwondo – while the South prefers Kukki-won Taekwondo.