The earliest Old Persian cuneiform we know of is in a stupid-huge trilingual inscription in Old Persian, Elamite cuneiform, and Babylonian cuneiform (basically well-aged Akkadian cuneiform). The inscription, at Behistun, Iran, is 15m by 25m, 100m up the side of a cliff face, and is basically Darius the Great bragging about how he totally kicked butt in various conflicts. Because it was so large and so inaccessible, the writing lasted quite well long enough to be deciphered. Stupid tank soldiers used it for stupid target practice in stupid WWII, and stupid damaged it. Fortunately, the damage was relatively mild, and happened after Old Persian had been deciphered.
Georg Friedrich Grotefend, building on the works of a few others, made some insightful observations/guesses in about 1835 that allowed the Old Persian writing system to start being deciphered, something that three others finished shortly afterward. It helped that Old Persian script was sort of an alphabet (with slanted wedge word dividers!) and that the language was relatively close to the well-known modern Persian.
Without the Behistun inscription, Old Persian cuneiform would not have been cracked so soon. Without Old Persian cuneiform, Elamite cuneiform would not have been cracked so soon or possibly at all, and Babylonian/Akkadian cuneiform would have been a lot harder.
Old Persian is not related to any of the other cuneiforms, nor obviously to any other language. Darius the Great claimed to have invented it, but he might have been exaggerating.
It is a somewhat unusual writing system to classify. It has syllables, and some of the syllables have an implied vowel. However, Old Persian also has vowels; the vowel can be added to a syllable to denote a lengthened vowel or a diphthong. Thirteen of the characters are for invariant consonants, i.e. do not change regardless of what comes after it. Finally, there are eight optional logograms. Thus it has aspects of an alphabet, a syllabary, and even of a logographic writing system.
Note that Old Persian, Babylonian, and Elamite cuneiform were still being used, 300ish years after the Latin script showed up, and 400ish years after Greek showed up. Cuneiform was a remarkably long-lived writing technology! Clay was cheap, paper had not been invented yet, papyrus only grew in Egypt, and cloth and leather were expensive.
Because so much was written on clay in antiquity, in antiquity history was frequently written by the losers. Cuneiform tablets were usually not fired on purpose, and so didn’t last. Many of the clay tablets which survived were ones that had been accidentally fired — when the city burned down, frequently at the hands of hostile invaders. Of course, sometimes fires would start by accident, but some of the richest archeological sites for fired clay tablets are from cities that lost big fights badly.