Like Herodotus, Thucydides etc.
What is the earliest extant historical source from Persia?
Your go-to first stop for all questions of Persian history is the Encyclopedia Iranica, which is a treasure, a collection of first rate scholarly articles on all aspects of Iranian history. Their essay "Historiography of the pre-Islamic Period" is required reading for anyone interested in early Persian history
It seems that their historical traditions were principally oral. They were _aware_ of Greek historians, and began to chronicle their own historical exploits, but not in book length. One of the first examples of Persians leaving us an historical record is the Cyrus cylinder but its a very limited bit of information, not a book length historical inquiry. There's no equivalent to Herodotus or Thucydides, at least that we know of.
Inscriptions and archaeology are the main historical sources we have for ancient Persia, and descriptions by the Greek historians-- we don't have a Persian historian like Herodotus, or an account like Xenophon's. The oldest Persian writing is mythological/religious, in the Avestan Yasts, which has some historical content, but isn't a "work of history"
For Persian history in a form recognizable as history, we have only Greek sources, Herodotus and a Greek physician who was at the Persian court, a prisoner of war named Ctesias. Ctesias wrote the first secular historical work devoted Persian history, the Persika; unfortunately most of this work is lost, all we have are fragments cited by other later writers.
One of the most interesting questions is: "Where did Herodotus get his information about the Persians?" -- because he had quite a lot of it. The answer seems to be from other people, who didn't themselves write books
The assumption that Herodotus had real and special sources of information as to Persian affairs, and the still more probable assumption that he did not find these when himself in the East, lead us to the conclusion that Herodotus must have met nearer home persons qualified to give him accurate and detailed information on Oriental matters, under circumstances which permitted him to question them carefully: such a source of information it is usually supposed that he found in Demaratus (cf. Matzat, Hermes vi. p. 479 seq., and others), who may well have furnished Hierodotus with many of his details as to Xerxes' invasion. The object of this paper is to suggest another and even more important source for his inner history of the Persian Court.
The passage in Herodotus is of considerable importance; he ends Bk. III. (c. 160) with the words 'the son of this Megabyzus was Zopyrus, who went over to Athens as a deserter from the Persians.'
The date of this desertion, and its significance will be considered later; first it is necessary to emphasize the fact that Zopyrus, if Herodotus really met him, is exactly the informant who satisfies the conditions of our inquiry, for he was one who was certainly able to give Herodotus the information desired, and one moreover who was likely to give it just in the form in which Herodotus reproduces
This "Zopyrus" sounds a bit like a Josephus (eg Josephus was a Jew who turns to Rome, Zopyrus seems to have been a Persians who turned to Greece), except rather than write a book himself, he's a source.
See "HERODOTUS THE HISTORIES AS A SOURCE FOR PERSIA AND PERSIANS" for an analysis of the strengths and problems of Herodotus Persian sources