This sounds like a question that could benefit from multiple perspectives, since its partially a question of institutional tradition. There is generally a sense in which Near Eastern history is more prevalent in ancient (western) history than in years past. This is particularly true at an undergraduate level. So a course on Ancient Greece might now spend some time discussing the internal mechanics of the Persian Empire, while a Late Antique course might devote a significant amount of time to the Sasanians. But this will vary a bit based on the institution. In a big university where Near Eastern history courses are taught, spending too much time on the Persians would mean stepping on a colleague's toes. It might well be that people studying Near Eastern empires aren't housed in Classics not because classicists don't appreciate their work, but because they've traditionally been in Archaeology, History, or Near Eastern Studies departments.

But on a broad level, there has a been trend towards expanding the horizons of classics. The principle movement here has been towards an appreciation of that the period from Constantine to the Islamic conquests is not one of simple decay and decline, but rather a continuation of classical antiquity with its own vibrant culture. This period is known as late antiquity, and it the man who largely invented it as field of study is Peter Brown. His 1981 Cult of the Saints is good, short, and gives a nice representation of his basic thought on late antiquity. If you're super interested, the first edition of the Journal of Late Antiquity has some good articles about the state of the field that are significantly more recent (late 2000's).

A second point of expansion has been the consideration of sources not in Greek or Latin. The use of demotic sources (demotic was was a language used in Egypt) by classicists to study Ptolemaic and Roman Egpyt has been underway for awhile now. A good recent example of such work is Joseph Manning's The Last Pharaohs (2009).

There is also an increased interest in using classical Arabic translations of works where the original Greek or Latin has been lost. Here the go-to book of the moment is probably Simon Swain's Economy, Family, and Society from Rome to Islam (2015), which is a discussion of Bryson's Management of the Estate, a tenth century Arabic source which is a translation of a Roman era guide to household management.

In addition to what has been said in this threat already, on my personal definition what is and what isn't Classics:

For instance, one of the PhD students at my college was studying the construction of royal authority in the Achaemenid empire; one of the reasons it was interesting, he said, is because classicists have largely ignored the Achaemenids except for their relationships with the Greeks.

"You know, Sanskrit is an ancient language with a body of classic literature in many ways similar to Greek or Latin; why is that not part of classics?"

These two questions are sort of related. I would say that the very rough traditional definition of Classics or History of Classical Antiquity is the study of the Greek and Latin speaking world and study of history from Latin and Greek sources. Of course this line is blurry. For example, I wouldn't consider a historian working solely on the Minoan civilization a classicist, perhaps a Mediterranean archeologist or something, but I could definitely imagine a classicist studying the development and influence of Minoan art on archaic Greek art. So, a classicist always studies things that are in relation to the Greek and Roman civilization. We could of course speculate, why does the study of very, very early Greek art, the Geometric period, make it so significantly more Greek than the Minoan culture? I guess it is easier to conceptualize an undisturbed line of development from the Geometric Athens to the Classical Athens, whereas Minoan culture sort of went dim around 1400BC and is not as obviously linked.

I'm a bit surprised that your friend is doing a PhD in Classics on the Achaemenid Empire. In my neatly compartmentalized world, I would classify that as Persian Studies or Iranian studies - Greek was never used by the Persians apart from as a language of diplomacy before Alexander the Great. Same goes for other ancient Mediterraenan/Middle Eastern cultures that were not Greek-speaking, people working on those are called egyptologists, assyriologists etc. etc. On the other hand, there's huge amount of ancient literature written in Latin and Greek about these cultures, so historians working on Achaemenid Empire will have to know Greek and Latin. But, you'll only get the Greek and Roman perspective, which is often orientalising and inaccurate, so what you're studying there is often really a history of the ideological relationship between the Greco-Romen world and the Achaemenid Empire. Which is the very reason why I find it weird that your friend is studying the Empire without any connection to Greece.

I recently did talk to someone who was puzzling over to what extent the classical languages should define what we study as classicists. He is working on the Hellenistic Susa, i.e. the era when it was under the Seleucid and Parthian empires, both of which had Greek as one of their official languages and they were politically structured along the Greek polis-model. It was, however, in many aspects a very Persian city as well: Persian and Aramaic were used for official purposes as well and 'indigenious' gods were worshipped alongside the 'Greek' pantheon. Around the end of the Hellenistic period, the city starts to slip further away from the Graeco-Roman world, and eventually they completely stop using Greek. This is also when Hellenistic historians, or classicists, traditionally stop working on Susa because the loss of the language is seen as the end of the Hellenistic era. But, my friend has been wondering, whether the language issue is too arbitrary a boundary. Susa still maintains its Greek political models and worships Zeus and other Greek gods long after it stops using Greek. So, is this justified?

Of course, Classical reception studies is a huge field of its own where people can study stuff like, the influence of Cicero on Donald Trump's rhetoric.