There have been a few iterations of the Persian empire. Did the successors see themselves as a continuation of the original empire, or were they successors only in culture?

I'll take this one since I'll be defending my master's thesis on the Parthian Empire in a month's time.

So, first we have to decide what qualifies as a Persian Empire. The first is obviously the Achaemenid Empire, ruling over the Iranian Plateau from the mid 6th century BCE to the mid fourth century BCE. After the fall of the Achaemenid Empire to Alexander the Great, the Seleucid successor state take over much of the old Persian Empire, though I don't think you could argue that they qualify as "Persian" though.

Then comes the Parthian Empire, founded and ruled by the Arsacid Dynasty, a semi-nomadic clan coming from central Asia that take over much of the territory of the old Persian Empire. The Arsacids couldn't ever claim ancestry to the Achaemenid kings, being so very clearly not-Iranian and so can't qualify as true "successors". The Parthians had a culture far different from the Achaemenids and their art and architecture reflect this.

The real problem, within my expertise, is how we classify the Sassanian Persians, who overthrow the Parthians in the 220s CE and make every attempt to claim they are rightful heirs of the original Persian legacy. The Sassanian royal line comes from Fars (where the Achaemenid Persians emerge from), thus being ethnically Persian, and they lose no opportunity in reminding the people they ruled over that they were the natural successors to the Achaemenid Persians.

The problem with this is that the Sassanians are less the successors to the first Persian Empire, as they are the successors to the Parthian Empire. As much as the Sassanians hated the Arsacid Parthians, when they took over, they only truly got rid of the royal family, and the entire noble family system of the Parthians remained as the governors and political elite of the new empire for several hundred years.

Beyond the Sassanian Empire, my expertise breaks down, but for these three Iranian Empires, the Parthian and Sassanians are far more related and connected than either the Parthians or Sassanians were to the original Achaemenid Empire, no matter how much they wanted people to believe it.

Interestingly, the one thing that is a pretty nice through-line in each empire is the dominance of Zoroastrian belief in each. The Achaemenid and Sassanian Persians made Zoroastrianism their primary faith, while the Parthians - in the style of nomadic and semi-nomadic groups - allowed the faith to flourish even if they never made it official imperial policy.

Some Sources:

Pierre Briant's From Cyrus to Alexander, 2009
Maria Brosius' The Persians: an Introduction, 2006
Malcolm Colledge's The Parthians, 1967
Neilson Debevoise's A Political History of Parthia, 1938