There was certainly continuity between the Arsacid and Sasanian Empires; you can arguably speak meaningfully of a single empire ruled by a succession of dynasties, although things like the shift in language makes it practical to separate them. Even toward the end of the Sasanian Empire, much of the power still lay in the hands of the seven "Great Houses" who claimed Arsacid heritage.

The continuity with the Achaemenid Empire is more complicated and less direct. The Arsacid Empire started as the province of Parthia around 250 BC in the Seleucid Empire, and was very Hellenic - for example, "the Philhellene" was a common epithet for kings. However, there was a gradual development generally traced to the time of Mihrdat I, who reigned from about 170 BC to 130 BC. Mihrdat was the great conqueror of the Parthians, taking what is today West Iran and Iraq from the Seleucids and the old Achaemenid cities. It is generally thought that discoveries such as the necropolis of Naqsh-e Rustam and its reliefs spurred a greater emphasis on the Iranian nature of the Empire, leading to such claims as Arshak, the founder of the empire, being a descendant of Artaxerxes I or II of the Achaemenid Empire, who were also called Arshak (or Arshaka, strictly speaking). It's unclear to when this claim dates, though. Mihrdat I was also the first Arsacid ruler to use the title Shahanshah, "king of kings".

What we know from later Iranian epic and Middle Persian writing is that the clear memory of the Achaemenids was essentially limited to Dara/Daryee (Darius III) and the invasion of Alaksandar-i Hromayig (Alexander the "Roman"), although there are signs of Artaxerxes I and II being vaguely remembered in the Shahnameh. Similarly, Ardashir and Shapur's claims to ancestry of diffuse, distant "forebears" who were clearly not the Parthians points to a cultural memory of a glorious past. Oral epic is difficult to use as evidence, since its ossification is often tied to the adoption of writing (hence why e.g. Vis and Ramin is set in 1st-century Parthia and the Shahnameh gets mythological beyond the Arsacids), but the preponderance of the evidence still points toward a lack of memory of Achaemenid rulers.

This shouldn't surprise us too much - after all, history is a genre of literature, which may simply not have existed in Achaemenid culture. Instead, they may well have preferred stories weaving myth and history together, something that is well-attested in Persian epic and Zoroastrian tradition.

However much "continuity" we can speak of, I don't think it should be understated that the establishment of Achaemenid ideology and administration was a basis on which later empires would thrive and improve. It was the greatest, most prosperous and powerful polity that had ever existed, and had an impact on its dominion and frontiers so massive we cannot even begin to quantify it.