The Medes built the Median Empire. The Parthians built the Parthian Empire. The Persians under the Sassanid dynasty built the Sassanian Empire. All of these peoples fall under the Iranian peoples denomination, who descended from Indo-Iranian language group speakers and inhabited the Iranian plateau.

Saying it belongs to the Kurdish history is bizarre. Kurds were part of these empires (not too sure about Median, someone could correct me). But they weren't Kurdish empires. They were all Empires dominated by Iranian peoples, which share some cultural heritage.

Almost all the empires in that region were of mixed Iranian origin. The Kurds are considered to be descendants of the Medes, so that might be why OP mentioned/asked it. Furthermore, these old languages (Parthian and Median) are directly related to the modern Kurdish language. Median and Parthian are both Northwestern Iranian languages. Median is Old Northwestern Iranian, Parthian is Middle Northwestern Iranian and Kurdish is directly derived from Parthian, namely Neo Northwestern Iranian. Many studies have been conducted on the origins of the Kurds. While not all historians agree with each other, many support the claim that the Kurds are the descendants of some of these ancient Iranian nations. Besides Kurdish, there is only 1 other Iranian language in the Northwestern branch, which is Balochi, but this language belongs to a subdivision of the Northwestern Iranian language (Northwestern II), and is not directly derived from Median or Parthian, like the Kurdish language is.

A lot of the ''Persian'' Empires were of mixed Iranian ancestry. For example, the first and best known ''Persian'' Empire, the Achaemenid Empire, was more Median than that it was ever Persian, yet people still refer to it as the ''Persian'' Empire (movies like 300 might have created that historically incorrect picture).

The first Emperor of the Achaemenid Empire (the movie 300 is based on a battle between the Achaemenid Empire & Greek states) was of Median royalty. Cyrus the Great's mother was a Median princess/queen, his grandfather was the Emperor of the Median Empire, and it was a Median general who brought him to power. The Achaemenid Empire adopted all the customs and ceremonies of the Medes, and many of its generals and politicians were Medes. Furthermore, the Achaemenid Empire was referred to as the Median Empire by many historians, Herodotus included, after the actual Median Empire fell.

Aren't Kurdish people a sub-branch of Persians?