According to Herodotus, there was no conquest in the sense. The Phoenician cities including Tyre had belonged to the Neo-Babylonian empire and recognized the suzerainty of the Persians willingly when Cyrus conquered Babylon in 539 BCE.
Specifically, in Histories 3.19 Herodotus says (in a different context, when talking about Cambyses, a later Persian ruler)
Cambyses did not think it right to apply force to compel the Phoenicians, both because they had delivered themselves over to the Persians of their own accord and because the whole naval force was dependent upon the Phoenicians.
It is also true that we do not have many extant sources about this time. In fact, our sources are basically limited to some Biblical scholars, who were not very concerned with Tyre; some Greek historians like Herodotus, and some scattered documents from Persian, Egyptian, or other sources. It is not that Persians and Egyptians (etc) did not write; it is more that later generations did not bother to preserve their writings as much as they did with Herodotus' books.
Katzenstein (1979) wrote an article about Tyre under Persian rule, in case you are interested in details. While this article is quite old already and possibly takes Biblical sources a bit too seriously, it does offer an in-depth view on the topic.
Edit (Apr 3 2020) on context and background: How did Tyre / Phoenicia come to be part of the Neo-Babylonian empire?
Whoever controlled the Levante (what is today Israel, Palestine, Libanon, and Syria) controlled the trade routes between the major empires in ancient Egypt, in Mesopotamia (Assyria, Babylon), and in Anatolia (the Hittites). As a consequence, these empires, whith which local powers in the Levante could generally not compete constantly fought for control over the Levante (e.g., here and here), except for a relatively brief period just after the Bronze Age collapse. At this time, Mesopotamia and Egypt were in turmoil themselves, while the Hittite empire collapsed and would never recover. This was when control over trade and commerce allowed Tyre and other Phoenician cities to become wealthy and powerful.
However, outside interference soon returned and subdued most of the local states that had sprung up in the meantime. With Tyre and the to some extent Sidon, this proved more difficult, as they were coastal cities and naval powers and able to keep supply routes open when they were besieged. The besieging empire would usually have to resort to devastating the surrounding countryside to discourage further resistance and force a compromise, usually a modest tribute paid by Tyre and nominal suzerainty of the empire. Over the centuries, Tyre was besieged or harrassed several times, including
These campaigns may have been costly and frustrating for the Mesopotamian attackers, but they were economically devastating for Tyre, as is detailed in the Wikipedia article.
Flavius Jesephus is our only source about Nabû-kudurri-uṣur's unsuccessful 13-year siege of Tyre. He relates that while the city was not taken, a compromise was concluded to the effect that Tyre paid tribute and recognized Neo-Babylonian suzerainty. And although the reliability of his account is disputed, there are other indications that Tyre was at least nominally a vassal of the Neo-Babylonian empire (such as the presence of the Tyrian king in Babylon as Nabû-kudurri-uṣur's "guest" (page 536-37 in this paper)).
