The birth of Vyāsa's son, Śuka, is narrated in the Mahābhārata as follows.
Bhishma said. 'The son of Satyavati having obtained this high boon from the great God, was one day employed in rubbing his sticks for making a fire. While thus engaged, the illustrious Rishi, O king, beheld the Apsara Ghritachi, who, in consequence of her energy, was then possessed of great beauty. Beholding the Apsara in those woods, the illustrious Rishi Vyasa, O Yudhishthira, became suddenly smitten with desire. The Apsara (Ghritachi), seeing the Rishi's heart troubled by desire, transformed herself into a she-parrot and came to that spot. Although he beheld the Apsara disguised in another form, the desire that had arisen in the Rishi's heart (without disappearing) spread itself over every part of his body. Summoning all his patience, the ascetic endeavoured to suppress that desire; with all his effort, however, Vyasa did not succeed in controlling his agitated mind. In consequence of the inevitability of what was to happen, the Rishi's heart was attracted by Ghritachi's fair form. He set himself more earnestly to the task of making a fire for suppressing his emotion, but in spite of all his efforts his vital seed came out. That best of regenerate ones, however, O king, continued to rub his stick without feeling any scruples for what had happened.
From the seed that fell, was born a son unto him, called Suka. In consequence of his circumstance attending his birth, he came to be called by name of Suka.
Based on the above story – which is rather perplexing to me – I'm unable to determine Śuka's birth-based varṇa. Can someone help me determine his varṇa?
Was he a brāhmaṇa by birth or did he belong to one of the lower varṇas? Finding out Vyāsa's and Ghṛtācī's birth-based varṇas will probably solve this problem. However, I cannot tell whether Ghṛtācī is a human or a parrot. For determining Śuka's varna, should I treat Ghṛtācī as a human or as a parrot? Or should I keep Ghṛtācī out of the equation because technically she had no role in Śuka's birth?
What does Manu-smṛti or any commentaries on it say anything about determining one's varṇa based on their father's varṇa alone? Also, does Manu or any other smṛti address special scenarios where the father of a child is human while the mother is a parrot?