The death date is a little unclear, might be better to say likely around 198 CE while Rafe De Crespigny prefers to date it around 199 CE

Parrot's don't turn up much in the texts I have gone through in terms of how they were kept. They were exotic creatures that Wu officer Xue Zong in 231 listed as a traditional example of exotic animal tribute from Jiao as he urged Sun Quan to treat the strange province with care. There is a story of Zhang Hua, a senior Jin statesman and scholar who pushed for the conquest of Wu, by Liu Song scholar Liu Jingshu. Zhang Hua had a white pet parrot who acted as a spy on the servants (and was put in a pot at one point to stop this so Zhang Hua ensured it was left outside) and even showed an ability to foresee the future via dreams.

So going on what the literary culture might tell us:

The expected pattern was "ah such magnificent bird, the colours, it is wise and can speak. Oh it is captured, it misses home a bit, so grateful to the kind treatment from it's lord and being in a place of culture". A message of my lord is such a generous figure, a man who draws the wisest to him, my lord has treated this unworthy scholar with such kindness and I am devoted.

Ni Heng as you said is noted for his Parrot Rhapsody but later in the court of the Han controller and warlord Cao Cao, the literary circles around his sons saw some compose on the theme as seemingly a joint challenge. Cao Cao's son and the most famed poet of the era Cao Zhi, the historian and poet Wang Can, Ying Chang who wrote a treaty on the art of surrounding chess, Chen Lin who is perhaps most known nowadays for his call to arms against Cao Cao for Yuan Shao and calligraphy expert Ruan Yu all composed on the theme of Parrot. All but Cao Zhi would be one of Cao Pi's Seven Masters.

Alas, I don't have access to the fragments of the seven masters parrot poetry that survive, I do have Ni Heng's and Cao Zhi's.

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Ni Heng:

While the use of birds for rhapsodies was not new, seems to have been the first on parrot or least the first that survived. When he wrote this, things had gone wrong in his life. He had gone to two of the great literary courts of the civil war, Cao Cao's imperial court and Liu Biao the Governor of Jing, and the brilliant but... troublesome scholar had ended up forced out of both. Now he was in the south of Jing, a place with its own culture, as a guest of Liu Biao's chief general in the region Huang Zu but with the patronage and support of son Huang Yi (sometimes translated as She). Though soon Ni Heng and the two Huang's would both be dead.

Someone brought a parrot to the party and Huang Yi (possibly to whom the parrot belonged in the first place) asked his famed scholar friend to compose something to entertain his guests, Ni Heng writing in one go and not entirely sticking to what might have been expected. Ni Heng's poem talks of this wise magnificent (beauty compared to a phoenix and goes into detail on colour) bird, from the western regions rather than the south, who can see the future and carefully chooses where it will land. Hearing of reputation and beauty, the court sends its best hunters to find it.

Captured, it chooses to avoid harm by acts of submission, by letting itself be touched though it is also trapped in a cage and its wings destroyed. It is forced miles and miles from home, relying on the kindness of others, unlikely to be boiled and eaten due to the smell of its flesh (though this doesn't seem to be true). It longs for home, worried about its fledglings and is separated from all it knows but keeps tame. Bemoans the times and that its talkative nature might be acting against it, it knows it is at the court of a distinguished gentleman but its tastes differ from the court and longs for home, it fears not having exceptional talent. It's mournful song during the autumn and winter months as it thinks of home moves those in similar situations.

Its broken wings mean it realises it can't go anywhere, growing bitter but so it serves its kindly master with all its heart for past kindness, putting its life and person in the lord hands and hoping to repay it with its own life and hoping the great favour of the master will not end.

Ni Heng does go for the usual beats, the colour, the intelligence, praising his patron Huang Yi and thanking for the kindness to this unworthy parrot, the new surroundings are impressive. But takes the usual "misses home" and goes quite deep with it, talking of how he is unsuited to the area despite the quality there, lingering on the longing for home and the mourning song, making clear he wants to go home and life here wasn't suiting him.

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Cao Zhi:

Written after Ni Heng, given Cao Zhi's age and that some of his fellow poets had yet to join Cao Cao while the Ye gatherings only begin around 208. Would probably have been before 212 when Ruan Yu passed away.

Cao Zhi's fu is more on the expected lines: speaks of a parrot as more beautiful than the finest birds of the central plains, touches on its migration to flee the cold to preserve its body. It is captured in a net, wings cut off and put in a cage, it's hen forced to return home without her partner. The parrot mourns not being able to see its children take wing but was expecting to be pulverized and boiled for eating. Grateful to its lord for generous treatment and of being cherished so seeks to add to lord's glory with it's, fears the kindness may sometime end but vows to keep toiling for one who has shown kindness far above the parrot's worth.

He plays with the expected themes, the majesty of the parrot (and the scholar), the trapped bird (who won't see the family growing up, which seems to be borrowing from Ni Heng's similar lines on this) but puts emphasis on gratitude towards the generous lord and seeking to add to "the lord's bright radiance", that he will keep working for his lord. In this case, his father Cao Cao.

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Overall:

Colourful, exotic, ability to talk (perhaps sometimes too much in troubled times), they were seen as smart creatures who could see the future and who would seek the worthy. They could be eaten, being boiled and chopped up, but the ones referenced to in literary circles were ones where their wings were broken and kept in cages for their ability to talk, colours and singing. Scholars could use these creatures to reflect on talent, to play a bit with the melancholy of the loss of freedom while being so grateful to their kind and worthy patron. Or in Ni Heng's case, a rather unusual "I'm grateful for everything, it isn't you but it is me, let me go"

On Jstor, I would recommend Mi Heng's "Rhapsody on a Parrot" by William T. Graham, Jr. on this. Great on the poetry including a copy of the poem, is dated on the politics that led Ni Heng to that situation but the poem and Graham Jr's analysis of themes including it's use of parrot should help.

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Sources

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Xue Zong and Wang Can's SGZ by Chen Shou with annotations by Pei Songzhi, translated by Yang Zhengyuan

The Poetry of Cao Zhi translated by Robert Joe Cutter

Chang Hua: A Statesman-Poet of the Western Chin Dynasty by Anna Straughair

Research on the Creation and Booming of Bird Fu in the Han Dynasty by Qunyi Ma and Shinian Ma

A Biographical Dictionary of Later Han to the Three Kingdoms 23–220 AD by Rafe De Crespigny