If by "Iranian" you mean "were Parthian and Khwarezmian Iranian Languages?", then yes. Taxonomicaly, Parthian is from the West Iranian Languages subgroup, so it's roughly speaking it would be more closely related to modern Kurdish, while Khwarezmian is from the Eastern Iranian Languages subgroup, and so is closer to the modern Pamiri languages or Pashto. Of course all Iranian languages also get classified into Old, Middle or New periods, and it's not 100% agreed how all of the taxonomy works across time and language.

As for "Persianized", I can speak a little better to Khwarezmians and the Khwarezmian Empire. Even in their core territory, inhabitants wouldn't have all been Khwarezmian speakers - a lot of the population was of Turkic origin at the time. The cities and especially the courts would have spoken Classical Persian, employed Persian speakers from elsewhere, and have heavily adopted Persian court culture.

Even in modern times, these sorts of linguistic and ethnic lines aren't so clear. Many people in Central Asia were bilingual Turkic and Persian speakers. Tajiks in what is now Uzbekistan and Tajikistan identified as the descendants of the ancient (Eastern Iranian-speaking) Sogdians, but speak a language (Tajik) that for all intents and purposes is a dialect of modern Persian. Similarly in Afghanistan, the two national languages are Pashto and Dari. Dari, which means "courtly", is also effectively the same as modern Persian.

This is good, I'll cover Parthia where u/Kochevnik81 dealt with Khwarezmia.

Parthia is a bit of a weird example. As the other comment notes, "Parthian" is a western Iranian language, which we really only know in its "Middle" form, though what we suspect to be Old and New Parthian are referenced in texts from other languages. This Parthian language is the common element in three originally distinct groups who were all ultimately called "Parthian" (or Parthian/Middle Persian Pahlavi ).

The original Parthians, the ones who were part of the Achaemenid Persian Empire, inhabited the area of northern Iran from about Tehran to modern western Afghanistan. This is where a theoretical "Old Parthian" would have been spoken. This region and its people were thoroughly integrated into the western Iranian empires of the Medes and Persians from the start (c. 600 BCE) and were incorporated into the Greco-Macedonian Seleucid Empire after Alexander the Great's death. Northern Iran, Parthia included, was actually more Hellenized and urbanized than the original Persian homeland in the south.

Over time, the Seleucid governors in Parthia became more and more independently until they seceded entirely. In 238 BCE, Parthia was invaded from the north by a steppe people called the Parni, ruled by a king called Arsaces I. This is group two and probably the one you were thinking of: the Arsacid Dynasty, often just called the Parthians by their Greek and Roman adversaries.

Arsaces and the Parni first conquered Parthia and gradually expanded from there over the next century, but it's not quite accurate to call them Persianized, at least initially. In reality the Parni were "Parthianized" and Hellenized. As Greek was the dominant political culture in the region, they adopted certain aspects of Hellenism, but they also adopted the local language and styles from the native Parthians. There were certainly Persian influences in both Hellenistic and Parthian culture, as well as all of the Arsacids' later conquests, but they were also more than a century away from the last time Persia held any significant power. They absorbed some of this influence, but it was contained within other more direct cultural contacts.

The Arsacids kept using the Parthian language as their own primary language for the rest of their time in power. At first it regularly appeared alongside Greek versions of the same text, and then Middle Persian translations in the late Arsacid Period once the province of Persia was gaining power.

We know basically nothing about the original culture and language of the Parni. They had previously been a part of the Dahae Confederation, a group of tribes on the eastern shore of the Caspian Sea. Depending on how different historians interpret different sources, they could have originated there or migrated from somewhere further east. Much later historians compared their language to the Scythians' Eastern Iranian language, but they are so much later that it's hard to trust them. There is a substrate of Eastern Iranian words in Parthian that probably originated in the Parni language and were still used after adopting Parthian.

The third and final group are the result of Arsacid influence. When they expanded their territory, the Arsacids made the city of Ecbatana (modern Hamadan) one of their capitals. After the Arsacid dynasty was overthrown by the Sassanid Persians, Persianization started to spread through the greater Iranian region, and the use of the word Parthia (Pahlava) started to shift. The original territory or Parthia where all of this started was increasingly included in the name "Khorasan" while "Parthia" was used to describe the area of Ecbatana. Early Arabic sources also refer to a language in that same region called "Pahlavi" (ie Parthian).

Whatever this final Parthian language was, by the Early Islamic Period, it was entirely divorced from the original region of Parthia and transplanted to the area known in antiquity as Media. Unfortunately, this language was too poorly documented to know if it died out or continued to evolve with a different local name.

By that time, centuries of Persian rule under the Sassanids had thoroughly imparted Persian culture across most of their empire. However, the Sassanids themselves were the products of about 500 years of Parthian rule and retained most of the Arsacid's political infrastructure. In a sense, the culture that Persianized the Parthians was itself already the Parthianized. Cultural and linguistic development is rarely linear