In short, mostly assimilated and integrated with the various different groups that passed through and settled in northeastern Iran over the centuries, like most ancient Iranian groups.

The Parthians specifically are an odd case. The ruling class of the so-called Parthian Empire, were actually cosmopolitan mixture of the Arsacid ruling family, descended from a steppe tribe called the Parni, nobility pulled from vassals across their Iranian territory, and inter-marriage with many of their neighbors, including Armenians and Greeks. The Parni conquered and subsequently integrated with Parthia in the 3rd century BCE, but the ruling class of Parthia was Greco-Macedonia, so the Arsacids and the rest of the Parni adopted a mixture of the indigenous Parthian culture and the Hellenistic ruling class that was in place when they arrived. They seem to have adopted the Parthian language for personal use and kept Greek around as an administrative and educational language until the end of their empire. After the Arsacids conquered the eastern parts of the Seleucid, various noble families took over the local administration, but many of these families were not actually Parthian/Parni/Arsacid, but Persian or some other group.

Part of the problem is that there really just aren't many for the Parthian Empire from the Parthian Empire. They existed in a gap between ancient clay and stone records and before medieval Islamic manuscript tradition, and thus many of their written documents went unpreserved. Their Sassanid successors did not do us any favors either. In many cases, Sassanid kings preferred the most politically or religiously expedient or impressive stories over the most historically accurate ones. This lead to significant erasure of the Parthian period in particular as Sassanid kings attributed events of the Parthian period to their own ancestors.

The Parthians also aren't known to have kept any written narrative histories in the modern Greco-Roman sense of the word. So the things lost are likely to have been administrative records, not stories. As a result we're left with coinage, royal inscriptions, engraved statues, artwork, and exceedingly rare fragments of parchment as our primary sources from within the Parthian Empire. To construct a detailed narrative sequence of events the Arsacid dynasty, we are almost entirely reliant on Roman Greek and Latin texts. As a result, we're very much bound by the Greco-Roman way of recording history, which was almost entirely focused on the rulers of a given country or kingdom. We can't even breakaway from that with the limited Parthian sources we do have, since they are almost all royal products.

As a result, it's almost impossible to describe any real history of the "Parthians" who actually populated the region of Parthia, beyond tracing their linguistic history through evidence of the Parthian language. In that regard, we can see that Parthia remained an important region and culture after the Arsacids were deposed by the Sassanids. In fact, the Parthian language may actually have been used more widely under the early Sassanids than the early Parthians. The former used it as a second language of royal inscriptions, but the latter primarily used Greek for official purposes.

After this point, beginning in the reign of Shapur I, things get more complicated witht e sudden appearance of a province called Abarshahr, apprently occupying the same location as Parthia had under the Arsacids, Seleucids, and Achaemenids, but also appearing alongside Parthia in royal inscriptions. Fast forward another 250 years and that same region is now part of Khorasan, one of the four Kusts instituded by Khosrow I in the 6th Century. The Kusts were overarching military districts that contained multiple provinces, but "Khorasan" became the standard identity for Parthia and the surrounding region until the modern period.

As Khorasan continued as a political unit, even after the Arab conquest, Persian became the dominant language, reflecting the Sassanids and later Persian-speaking dyansties that ruled it from the south. The earlier provincial identities, like Parthia or Abarshahr were gradually forgotten as Khorasan became the more prominent regional identity. However, it was also on the frontline of basically every incursion from the steppe to enter Iran. In the late-Sassanid and early-Islamic periods, that was the Hephthalites, but over the last 1300 years it has included many Turkic and Mongol groups, including the Seljuks, Ghengis Khan, and Timur, amongst others. Each successive group influenced and mixed with the people of Khorasan, including the descendants of the original Parthians. As a frontier, the region has also seen significant settlement from groups sent to garrison or serve an exile in the region as well, ranging from Arabs to Kurds to Balochs. In all of these cases, the modern groups identifying as Turks, Timuri, Arabs, etc, are descended from a combination of these outsiders moving in and locals assimilating into those cultures.

Under the Sassanids, "Parthia" apparently came to refer to an area in the northwest, between modern Tehran and the border with Iraq, which now encompasses Iranian Azerbaijan, Kurdistan, Hamadan, Kermanshah, etc. Based on early Arabic references to "Parthia," this is seemingly the same region identified as "Jibal" in the early Islamic Period, and Media in the Achaemenid Period. This region was seemingly identified as the core of Arsacid Parthian culture, as it hosted many of their most prominent cities, like Hamadan and Ray. Hamadan (ie Ecbatana) was probably one of the most important factors to this shift. The later Arsacids made the former Median and Achaemenid fortress city into one of their main capitals, and the wider region seems to have become associated with them.

The redefined region of "Parthia" centered on modern Hamadan was identified as Jibal until the 13th century, but early on it was sometimes grouped together with Azerbaijan as "Pahla" or "Fahla". This name is the root of several words that can be confusing if not properly distinguished, but is also probably a clue to what happened to the last "Parthians." In modern scholarship "Pahlavi" is often used as an alternate name for the Middle Persian language, but that name stems from Middle Persian's use of a "pahlavi script." This use of "pahlavi" originally referred to the family of Iranian alphabets derived from Aramaic during the Arsacid period, in contrast with modern Persian/Farsi use of an Arabic derrived script. It was first used for Parthian before being adopted for use with Middle Persian and Avestan.

The Arabic name Fahla was thus the ultimate, form of "Parthia" (Old Persian Parthava > Middle Persian Pahlav). In the Islamic Period "Fahlaviyat" was a genre of poetry composed in the local dialects of the Fahla region, but by the 11th Century these were considered dialects of Persian rather than a distinct Fahlav (ie Parthian) language. However, their position as distinct dialects points to some kind of additional influence, which was likely a form of Parthian in some cases. Despite this, Persian ultimately became the dominant language and cultural influence in the region. The process began under the late Sassanids and blended with Arabic influence after the Arab conquest, and after the Arab occupation neither Parthian or regional Fahlav dialects were ever used again in an official capacity, with Arabic and Middle or Modern Persian becoming the administrative and day to day languages of all subsequent rulers.

So there you have it, the original Parthia was ultimately absorbed into the mulitcultural milieu of medieval Khorasan, while the later region that adopted the name was at the intersection of Persian and Arabic culture. That said, there is one last legacy of "Pahlav" that should be mentioned. If you're at all familiar with recent Iranian history, then you may know "Pahlavi" better as the name of Iran's last monarchical dynasty. Their name is evidently derived from the later name for Parthia.

Reza Shah Pahlavi was born in the Mazandaran province, which sits along the south shore of the Caspian Sea just north of the region historically identified as Fahlav or Jibal. Mazandarani is a extant language today, but seems to be descended from other languages spoken around the Caspian rather than ancient Parthian. However, as it sits directly north of Fahlav, it stands to reason that the name "Pahlavi" originated to denote a family from Fahalv/Pahlavi, and by very long extension: Parthia.