To my utter shock, I don't have a canned answer to link you to, but I have written about this topic broadly speaking as part of a lot of other answers on here and will pull from those. So: apologies if anybody has deja vu reading this.

The Sassanids, especially early on, had a crystal clear understanding of the Parthian Empire because there was no break in continuity. Ardashir I simply rebelled and took over the Parthian state. If you're already familiar with Achaemenid history, its fairly similar to Cyrus the Great rebelling against Astyages and taking over the Median "Empire." Historian Parvaneh Pourshariati even calls the Sassanid Empire a "Sasanian-Parthian Confederacy."

Over time, the finer details of Parthian history fell out of Sassanid memory, partly as a result of of Zoroastrianism's millenarian eschatology. By the late Sassanid period, and perhaps earlier, Zoroastrians developed a belief that time moved in cycles with a major divine shift occurring every 1000 years, though the actual phrase hazagrok ziman is vaguer, more like "1000 ages." The revelation of the prophet Zarathustra marked the beginning of the "current" millennium, and thus the millennium could not end until some great divine shift occurred, commonly interpreted as the beginning of Frashokereti, the Zoroastrian apocalypse.

One of the few surviving early Sassanid documents, The Letter of Tansar, suggests that Ardashir's contemporaries saw the end of time as imminent. The high priest Tansar wrote:

This gate to goodness and good order, set open by [Ardashir] for the people, will remain so for a thousand years; and were it not, as we know, that after a thousand years, by reason of neglect of his testament, riot and disorder will come into the world, and that all that he bound will be loosed and all that he loosed will be bound, we should say that he had toiled for the world to eternity.

When Frashokereti did not come, obviously the count of the millennia had to be reassessed. Unfortunately, we don't know what exactly this looked like because many Sassanid sources have been lost, leaving us to rely on later medieval Zoroastrian texts that cite them. By then, the count of millennia had already been revised again, apparently by shifting how many millennia had already occurred rather than by further compressing their timeline. In regard to Parthian history, this manifests in the commonly repeated statement that the Arsakid Parthian dynasty ruled for fewer than 300 years from the death of Alexander the Great (Alexander the Accursed in Zoroastrian texts). Exact numbers vary from text to text, but all essentially remove the first 250 years of Parthian time, and the preceding 100 years of Hellenistic rule, from the timeline.

However, it's not like the Sassanids just started history with the historical Parthian kings from 60 BCE onward. They still remembered the likes of Arsakes I and other significant rulers from the early period, but compressed them all into the ~280 years allotted to Arsakid rule and didn't question exact numbers much in the surviving sources.

That said, they certainly seem to have forgotten all about most of the Achaemenids and Seleukids. This is tied to both millenarian theology and loss of records. They remembered that Alexander invaded and conquered a king called Darius, Middle Persian Dara and that this was not the first king of that name, attributing Alexander's wars to Dara II, son of Dara I. Of course, that's not inline with any Achaemenid or western records. Alexander defeated Darius III, son of Arsames and distant descendant of Darius II (and through him Darius I the Great).

The Dara of later Persian and Zoroastrian tradition was the last of the Kayanian kings, taking their dynastic label from a title used in the Avesta for many great heroes, who are not all presented as monarchs in the scripture. The likes of Cyrus the Great, Xerxes, and all of the Artaxerxes are completely absent from this later tradition. Darius the Great alone may have had some extremely indirect influences on the legends of other characters, namely: Jamshid, Fereydun, and Goshtasp. Ironically, these figures of legend were already present in the Avesta, which was largely composed before the Achaemenid period, and earlier forms of their stories were probably familiar to the Achaemenid kings. In the Sassanid reckoning of history, they were organized into a single dynasty of rulers who occupied the same place in history as the Achaemenids as the rulers who preceded Alexander.

Millenarianism mandated that Zarathustra mark the beginning of the current millennium. So Zarathustra and his patron Goshtasp were placed 553 years before Ardashir. Of course, in real history this only goes back to 329 BCE, contemporary with Alexander, but remember we've already deleted about 280 years. This places Zarathustra in the functional equivalent of c. 600 BCE while allowing the Sassanids to make use of a slightly modified version of the popular Seleukid Era calendar as the Era of Zarathustra instead.

It also placed Goshtasp in the historical position of a real individual of the same name. Goshtasp is the Middle Persian cognated of Greek Hystaspes and Avestan Vishtaspa, the name of Darius the Great's father. Presumably, the Achaemenid Hystaspes was named after Zarathustra's patron, but the 4th Century Roman historian, Ammianus Marcellinus, identified them as one and the same. Whether this co-identification caused Goshtasp to be placed 553 years before Ardashir or the Zoroastrian date influenced Marcellinus is impossible to say. In reality, modern analysis of Zarathustra's Gathas places the prophet around 1200 BCE, something supported by Classical Greek authors' consistent belief the Zarathustra was already incredibly ancient in their own time, but that would obviously upend Sassanid millenarian beliefs.

The Letter of Tansar also sheds light on how the Achaemenids were forgotten. Tansar wrote:

Yet have you heard tell of, or seen, any monarch save the King of kings, who has taken this task upon him? With the vanishing of religion you have lost also the knowledge of genealogies and histories and lives of great men, which you have let pass from memory. Some of it you have recorded in books, some upon stones and walls, until none of you remembers what happened in the days of his father. How then can you recall the affairs of the people at large and the lives of kings and above all the knowledge of religion, which ends only with the end of the world?

Some of this is a reference to the loss of written documents and knowledge of how to read the Achaemenid cuneiform monuments. Some of it is also just Tansar complaining about the poor state of oral memorization of the Avesta in 3rd Century Iran, which ultimately led to the Sassanids writing it down. The Sassanid era Persians knew that they had forgotten their own history, and Tansar openly lamented it. Later generations worked backward from what they had, taking the treasured stories of oral tradition and religious figures to insert them into history. There is a clear memory across medieval Zoroastrianism, Sassanid histories, the Shahnameh, and even late-Roman histories that the Persians remembered that their ancestors once controlled an empire that stretched much further west, that Persepolis was the capital of a great king (the Throne of Jamshid in later tradition), and that it all fell apart because of Alexander. They simply identified this as the Kayanian period rather than the Achaemenid.

Why they didn't just read the very detailed Greek histories is unclear. Maybe they did and dismissed them because they were so out of line with Zoroastrian doctrine, or they read them and directly interpreted the Achaemenids as Greek fictions about the Kayanians. Ardashir, and to a greater degree his successors under the influence of the powerful priest Kartir, did make a concerted effort to assert Iranian identity in contrast with the Greco-Roman west. That included eventually abandoning Greek in monumental inscriptions (though Ardashir himself did use Greek alongside Middle Persian and Parthian at Naqsh-e Rostam). However, they were still in regular contact with Rome. There were Sassanid scholars and diplomats perfectly capable of reading Greek, and Greek scholars came to Ctesiphon and the great library of Sarouyeh. The apparent lack of Greek influence on surviving Sassanid historical tradition is still a mystery.