As you note, Parshandatha was the governor over קרדוניא which Jastrow's dictionary (p. 1412) translates as:
Kardunia, Coryene, a district lying to the east of the river Tigris, south of Armenia.
Today this historic region is often referred to as Corduene, and at various points in time was considered Armenian. There is an informative Wikipedia entry on it: Corduene
The Targum Yonathan (Bereshith 8:4) states that it was in Corduene/Armenia that Noah's ark came to finally rest:
וּנְחַת תֵּיבוּתָא בְּיַרְחָא שְׁבִיעָאָה הוּא יַרְחָא דְנִיסָן
בְּשֶׁבְסְרֵי יוֹמִין לְיַרְחָא עַל טַוְורֵי דְקַדְרוֹן שׁוּם טַוְורָא
חַד קַרְדַנְיָא וְשׁוּם טַוְורָא חַד אַרְמִינְיָא וְתַּמָן
מִתְבַּנְיָא קַרְתָּא דְאַרְמַנְיָא בְּאַרְעָא מַדִינְחָא
And the ark rested in the seventh month, which is the month of Nisan, in the seventeenth day of the month, upon the mountains of Qadron; the name of the one mountain is Qardania, and the name of the other mountain Armenia; and there was built the city of Armenia in the land of the east.
These are presumably what are known today as the twin peaks of Mt. Ararat in Armenia.
BT Sanhedrin 96a reports that at one point Sanheriv had found a plank from Noah's ark and reasoned that it was a god named Nisrokh:
אזל אשכח דפא מתיבותא דנח אמר היינו אלהא רבא דשיזביה לנח מטופנא אמר אי
אזיל ההוא גברא ומצלח מקרב להו לתרין בנוהי קמך שמעו בנוהי וקטלוהו היינו
דכתיב (מלכים ב יט, לז) ויהי הוא משתחוה בית נסרוך אלהיו ואדרמלך ושראצר
בניו הכהו בחרב וגו'
He [Sennacherib] went away and found a plank of Noah's ark. 'This',
said he, 'must be the great God who saved Noah from the flood. If I go
[to battle] and am successful, I will sacrifice my two sons to thee',
he vowed. But his sons heard this, so they killed him, as it is
written, And it came to pass, as he was worshiping in the house of
Nisroch his god, that Adrammelech and Sharezer his sons smote him with
the sword etc.
This would seem to introduce a possible contradiction concerning the location of the wood as Sanheriv had removed it quite some time prior to the events of Purim. However 2 Kings 19:37 records that these two sons then fled to Ararat:
וַיְהִי הוּא מִשְׁתַּחֲוֶה בֵּית נִסְרֹךְ אֱלֹהָיו, וְאַדְרַמֶּלֶךְ
וְשַׂרְאֶצֶר (בָּנָיו) הִכֻּהוּ בַחֶרֶב, וְהֵמָּה נִמְלְטוּ,
אֶרֶץ אֲרָרָט; וַיִּמְלֹךְ אֵסַר-חַדֹּן בְּנוֹ, תַּחְתָּיו
And it came to pass, as he was worshipping in the house of Nisroch his
god, that Adrammelech and Sarezer his sons smote him with the sword;
and they escaped into the land of Ararat. And Esarhaddon his son
reigned in his stead.
And based on this, the Rav Pealim (R. Avraham b. HaGR"A) as reported in She'arim Mesuyanim (p.270) suggested that Sanheriv's sons brought the plank of wood with them to Ararat, thus furnishing the opportunity for Parshandatha to obtain it when he was the governor there.
The Targum Sheni 9:24 contends that despite the biblical injunction against leaving a body hanging over night (Deut. 21:23), Esther maintained (by reference to the hanging of Saul’s sons as precedent) it was appropriate to leave the bodies of Haman and his sons up on the gallows not just for six months, but forever:
מטול דהמן בר המדתא אגגיא מעיקיהון דיהודאי חשיב על יהודאי למגרמא יתהון
צבע פיסא רמא ערבא וסבתמא לכל עדבא למגרמא יתהון ולמשיצאה יתהון. וכד חזו
להמן ולבנוי דצליבן על צליבא יומין סגיאין אמרין למה אסתר עברה על מה
דכתיב לא תבית נבילתיה על צליבא ענת אסתר ואמרת להון על דקטל שאול מלכא
לגיורא דגבעונאי הוון בנוי צליבין על צליבא מן שירוי דחצדא דשערין עד
יומא דנחת מיטרא דהוון שיתא ירחין. וכד הוון דבית ישראל סלקין נתחזון
בבית מקדשא הוון עממיא אמרין להון למה אילין צליבן ענו דבית ישראל ואמרין
להון על דפשט אבוהון ידיה בגיורי דגבעונאי וקטל יתהון. המן רשיעא דבעא
לשיציא לדבית ישראל על אחת כמה וכמה יהוי הוא ובנוי צליבן על צליבא כל
יומי עלמא.
When the royal scribes of Haman saw him and his sons hanging many days
on the gallows, they asked: "Why does Esther transgress what is
written in the law, ‘Thou shalt not let his body remain all night upon
the tree’? Esther answered them: "Because King Saul had killed the
proselytes of the Gibeonites, his sons hung upon the gallows from the
beginning of barley harvest until the day when rain came down upon
them, which lasted six months; and when the Israelites came up to
appear before the temple, the nations asked them: 'Why do these hang?'
The Israelites answered them: 'Because their father has laid his hand
upon the proselytes of the Gibeonites and killed them.' How much more
does the wicked Haman and his sons, who wanted to destroy all Israel,
deserve to hang on the gallows — FOREVER!"
Accordingly, it may be presumed that the gallows stood at least as long as Esther reigned. Furthermore, its use as such may be argued was a perpetuation of the original function for which the wood served. She'arim Mesuyanim (p.270):
יוצא לן מחלוקת האם הזכר עשה לנפלאותיו הוא לזכור הצלת העולם חסדו של
הקב"ה או לזכור גבורת השי"ת שהשחית את העולם ונ"מ בזה מה שנתלה המן על עץ
זה. דאם העץ הוא זכר לגבורות השי"ת במבול שהשחית את הרשעים מסתברא שהמן
ג"כ נתלה בשביל רשעתה משא"כ אם הוא זכר לחסדי השי"ת במבול א"כ גם תליית
המן הוא המשך לנפלאות השי"ת שהציל את העולם מן המבול וגם הציל בו את
ישראל מגזירת המן
What emerges from the debate is whether the “remembrance of His
wonders” is a remembrance of the loving-kindness (hesed) of God [in
His abating of the flood], or as a remembrance of the strength
(gevurah) of God when he destroyed the world. This distinction also
determines how we see the hanging of Haman upon the wood. For if the
wood was a remembrance of the might (gevurah) of God who destroyed the
wicked with the flood, it is evident that Haman similarly was hanged
because of his wickedness - which is not the case if it was a
remembrance of the loving-kindness (hesed) of God. However, if the
latter [i.e. that it is a remembrance of the loving-kindness of God]
is in fact so, then we see that hanging of Haman is also a
continuation of “His wonders” for just as He saved humanity from the
flood so too did He save the Jewish nation from Haman’s decree.
And while we may not know what happened to the gallows consequent to Esther's reign, it would seem that according to some sources the remainder of the ark continued to serve as a memorial in its original location. Josephus (Antiquities of the Jews, Book 1, ch. 3) reported:
the Armenians call this place, 'The Place of Descent'; for the ark
being saved in that place, its remains are shown there by the
inhabitants to this day.
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