Regarding Knights... well successor or follower... what I meant to say is that the Sassanians had a concept of Knighthood (savaran / asvaran) with many aspects (a caste of warrior-nobles, with advanced full body armor, with a code of conduct, chivalry, lance duels etc.) that found westwards over time it's way to the high medieval European knighthood. The Persian heavy armory (clibanarii, cataphracts) and many other military concepts and tactics was copied one-to-one by the late Romans and is the predecessor for medieval armor, the heavy cavalry merely being the most prominent.
Also ideological influences (not all being exactly positive) from Persia had a lasting influence on the Romans. Well known is the cult of Mithras which was widely popular among Roman soldiers. But also the very concept of sacred godlike Emperorship, was adopted from the Persians by Alexander and through the following Hellenistic Kingdoms found it's way to the Romans (who were better served with a Republic). During the Republican time, as Plutarch reports Persian priests had an influence on Lucius Cornelius Sulla (when fighting in Asia), and led him to believe himself to be a deity. The later Roman "divine" Emperorship was clearly an Eastern concept. Also "Augustus" is derived from the Avestic "aojishta" (possessor of the increasing forces).
George Lenczowski's book on Imperial Iran has a chapter on "Sacred Kingship", which covers numerous details of these influences on Roman Caesars.
The Persian Monarchs were never worshipped as divine by the priests of Ahura-Mazda as they were strict monotheists. Alexander borrowed the Court protocol of the Persians which was probably borrowed from Mesopotamians. As for the term "Augustus" being derived from Persian, I think that is a stretch. Both Latin and Persian are Indo-European and share certain root words. It is more probable that they developed independently from a common source.