I

I'll address all of this eventually but first, one thing stuck out to me in your question:

even when they used Persian, they often preferred to use cuneiform or the strange Pahlavi script

and

Why did Persia seem to neglect their native Persian language for written communication?

These are contradictory statements. What would you have them use?

Old Persian Cuneiform and Middle Persian Pahlavi were scripts adapted in a unique way to write the Persian language. It's roughly the same concept as the adoption of Arabic script, or the development of the Latin or Cyrilic alphabets from Greek to better represent Italic and Slavic languages, respectively. Reaching further back, the same thing could be observed about the Greeks, Arameans, or Hebrews adopting the Phoenician alphabet to their own use or even the Phoenicians using a variant of the Proto-Sinaitic Egyptian script.

Just as the medieval Persians adopted an Arabic style script as the result of Arabic influence, the Achaemenids developed a cuneiform-style alphabet under the influence of Elamite and Akkadian as the hegemonic languages of their day and the Sassanid Persians used Pahlavi after the Parthian hegemony of their day. The Pahlavi script in turn was developed by adapting Aramaic another widespread language that used both inside and outside of Persia.

Still, there remains a question of why other languages were used for administrative purposes. The basic reasoning is that the Hellenistic and Roman method of using one language for as much as possible is actually fairly inefficient, especially at first. It requires the mass education of subjected peoples in the new language, at least in governance and commerce. However, if there is already a widely accepted language used in most of the region you're conquering the only reason not to make use of that is pride. That was the case for the pre-Islamic Iranian dynasties.

The Achaemenids made use of four primary languages (and then regional use of local tongues). Aramaic was the most widely used language in the empire because it was already the most widely used language from Afghanistan to Egypt when the Persians rose to power. It had risen to international prominence under the Neo-Assyrian Empire both because it was widespread and because the Aramaic alphabet was easier to learn than Akkadian cuneiform. However, Akkadian was also used in the Achaemenid Empire, but mostly as the local language of Babylon and a prestigious language for monuments. It was the language of learning and government for millennia from the Mediterranean to the Zagros Mountains, and that carried prestige with it, much like using Latin and Ancient Greek does today.

For most Achaemenid history, Elamite held a similar role to Akkadian, but its historical prestige came from being the historic language of the Persian homeland, which had previously been the Elamite region of Anshan. The early Persian kings from Cyrus to Xerxes apparently used Elamite as an administrative language for the regional administration in southwestern Iran around the palace capitals at Susa, Persepolis, and Pasargadae, but it was ultimately superceded by Aramaic for the reasons listed above. Much like Akkadian, Elamite Cuneiform was also just harder to learn and memorize than the Aramaic Alphabet.

Old Persian is the last major language, and was probably almost exclusively written on monuments. As the language of the Persians themselves it was the spoken language of the ruling class, but was both a minority tongue and somewhat nouveau riche. Prior to Cyrus the Great's conquests Persia was an extremely minor kingdom in a disjointed Median Empire/Confederacy. It had no international spread. Since nobody but the Persians and maybe the Medes really knew Persian it didn't make much sense to try and spread it when there was already a perfectly good lingua franca in the form of Aramaic. On top of that, Old Persian was not written down until the reign of Darius who may have commissioned the cuneiform-style script specially to match the Akkadian and Elamite he was using on his monuments, but in an easier to learn format.

Then everything changed when the Macedonians attacked. The Hellenistic Period ushered in a Greek ruling class with Greek sociological and philosophical notions of how to govern. Part of that Hellenistic outlook was to use the Greek language. This meant two things for the language used in Persia and the former Persian empire. Greek became the primary language of government record keeping and communication. It joined Aramaic for commerce and local administration continued, especially in Mesopotamia and Syria. It also meant that Old Persian cuneiform, to the limited degree that it was ever used and taught among the nobility, was no longer being taught. Persian writing disappears from Persian and Median tombs and monuments in this period. It seems that Persian cuneiform was only ever supported and spread by the Achaemenid dynasty while day to day business was done in more widespread languages.

Much like the Achaemenids, the Parthians conquered territory while speaking a language that had no associated writing system. They conquered a kingdom with a Greek speaking government, and much like the Achaemenids had initially use Elamite, they simply adopted the administrative apparatus they found in place. Greek had none of the educational baggage of traditional cuneiform and most of the same international utility as Aramaic at that point, so it stuck around for the duration of the Arsacid Parthian dynasty. At the same time, Parthian was the spoken language of the ruling class, and much like the Achaemenids they developed a script to suit it for monumental and communications purposes. This was the origin of Pahlavi writing. Pahlavi is just an Iranian word for Parthian. Meanwhile, Aramaic just kept going along, though the rise of Greek kingdoms in both the east and western sides of the Aramaic world had given it a competitor.