Yay, I question I can answer!

This is a very complicated topic, and one that needs to be explained on multiple levels.

For starters, let us start with the fact that Achaemenids considered themselves successors to the Assyrian, Babylonian, and Median empires and adopted many of their titles (king of kings for instance). After Alexander defeated Darios III, things changed; the empire was now under the control of a Makedonian. But, it seems that the sources do emphasize that Alexander Persianized to a degree like adopting their clothes and the practice of proskynesis. There is also Alexander burying Darios III; the sources all say that this was because he was such a nice guy (news flash: he wasn't). Instead, it plays to a succession motif where the son buries the father and thus assumes control of the household, or in this case the empire. At a later period, the Parthians under king Phraates II would do the same and bury a Seleukid king.

After this death, things get even wonkier. There is no specific evidence to suggest any degree of hostility of the Greco-Makedonian successors and the Persians (or Iranians in general); in fact there are more cases of cooperation with the example of Antigonos Monophthalmos having his Persian dinner with Persian nobles in the front (an important gesture).

Now, when we enter the Seleukid period, we notice a few things. Now first off, while rather controversial, I agree with the sentiment that the Seleukids were indeed a Persianate dynasty in that they adopted many elements of Achaemenid rule like the satrapal system and court procedure. Even better, Seleukos' marriage to Apama was symbolic as it was the only marriage of Alexander's generals to Persian wives to survive; Antiochos I was by definition a half-Persian, showcasing the physical union of two culture groups. There is also evidence of the Seleukid armies employing Persian contingents and a bodyguard of Median cavalry with the king.

There is little else to suggest a full on Seleukid willingness to be called successors to the Achaemenids. There have been excavations into Seleukid inscriptions at Behistun which shows Herakles. Historians are divided with this relief, because some argue it was a show of Hellenic imperialism to the Iranians; more likely it was a symbol to showcase who the Seleukids were, and there have been works that translate Hellenic Herakles with the Iranian deity Verethraghna, so the image catered to different audiences.

For the most part, histiographically, those that argue for the Seleukids to be a continuation of the Achaemenids are Amelie Khurt and Susan Sherwin-White in their book "From Samarkhand to Sardes." It is a good book, but can get a little preachy at times.

Now come the Parthians, and these guys get even weirder. A lot of people see them as a continuation of the Persians, but that is a very dangerous statement to make. For starters, the Parthians were a nomadic group that lived near modern day Turkmenistan, and in 247 BC moved into the modern Iranian provinces of Semnan and North Khorosan. They would remain here until the 170s BC when they moved and conquered up through Mesopotamia. But even in this period, it is unclear how Persianized the Parthians were; instead a more apt question is how Hellenized they were. Yes, it appears archaeologically the Parthians adopted many Hellenic elements into their culture; BUT this was never done to showcase how awesome Greek culture was, rather they fused both Hellenic and nomadic elements to make their own distinct culture. By this period, there is no evidence to suggest that the Parthians were even remotely interested in the Achaemenids of old. From what we can glean, a more apt characterization is to make the Parthians less successors to the Achaemenids than successors to the Seleukids (For more, look into the excavations into Nisa).

Things start to change with the king Mithridates II. If you look at some of his coins (particularly the Sellwood type 29, all variants), he has adopted the title "King of Kings" and has a fancy tiara. Numismatists (like Dr. Vesta-Curtis) argue that this is the first element of the Persianization of Parthian ideology. However, it must be noted that the title "King of Kings" is written in Greek which is an important point, and may suggest a retention of Hellenistic elements in their culture.

In the ADs (sorry this isn't really my specialty) there is a telling quote by Artabanos II as narrated by Tacitus: The Parthian king asked for the return of borders held by Alexander the Great and Cyrus the Great. Professor Rahim Shayegan has gone into this and argues that it was a response to Roman imperialism in Asia Minor that induced the Parthians do adopt pro-Achaemenid rhetoric as a diplomatic tool to create the boundary for their empire as opposition to the Romans. (For more, please read Dr. Shayegan's book: Arsacids and Sassanians, it is a great book, I highly recommend it.) With regards to the Romans, it seems that their sources attempted to recreate the past conflict of Greece vs Persia, and thus the Romans were turned into the heroic freedom party, while the Parthians became the despotic evil empire. Naturally, this is propaganda, but because of this trope, some sources do like to make the Parthian into basically the Persian, even though there is little literal evidence of this being the case.

With the Sassanians (now I am really out of my element), Dr. Shayegan argues that an initial attempt was made to turn the Sassanians into the Achaemenids of old, but upon doing really well militarily (at one point, they humiliated two Roman emperors), they still referred to them as Persians, but a more bellicose and troublesome enemy than the Parthians. Critically, he argues that the Roman sources made the Sassanians as heirs to the Achaemenid empire to allow for this trope to function, so the idea of the Sassanians being the "real" successors to the Achaemenids was likely a myth, or at least an amplified version of what we see. BTW, I cannot find the source, but there have been arguments that the Sassanian clan were themselves descendant of a Parthian clan for Persis, and took their geography to proclaim a reformed Persia.

Well I hope that answers your question, and if you have anymore, please do not hesitate to ask!

Sources: Primary-- Arrian, Anabasis of Alexander the Great

Diodoros World History

Herodotos, Histories

Justin Epitome of Pompeius Trogus

Plutarch, Life of Alexander

Secondary: "Age of the Parthians" by Vesta Sarkhosh Curtis

"Arsakids and Sassanians" by Rahim Shayegan

"From Samarkhand to Sardes" by Amelie Khurt and Susan Sherwin-White

Encylopedia Iranica

parthia.com