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Śrīmad-Bhāgavatam 1.4.14

సూత ఉవాచ ద్వాపరే సమనుప్రాప్తే తృతీయే యుగపర్యయే । జాత: పరాశరాద్యోగీ వాసవ్యాం కలయా హరే: ॥ ౧౪ ॥
sūta uvāca
[soota uvaacha]
dvāpare samanuprāpte
[dvaapare samanupraapte]
tṛtīye yuga-paryaye
[triteeye yuga-paryaye]
jātaḥ parāśarād yogī
[jaatah paraasharaad yogee]
vāsavyāṁ kalayā hareḥ
[vaasavyaam kalayaa hareh]
Word-for-word
sūtaḥ — Sūta Gosvāmī uvāca — said dvāpare — in the second millennium samanuprāpte — on the advent of tṛtīye — third yuga — millennium paryaye — in the place of jātaḥ — was begotten parāśarāt — by Parāśara yogī — the great sage vāsavyām — in the womb of the daughter of Vasu kalayā — in the plenary portion hareḥ — of the Personality of Godhead.
Blended reading (Sanskrit + English inline)
sūta uvāca
sūtaḥ
Sūta Gosvāmī
uvāca
said
dvāpare samanuprāpte
dvāpare
in the second millennium
samanuprāpte
on the advent of
tṛtīye yuga-paryaye
tṛtīye
third
yuga
millennium
paryaye
in the place of
jātaḥ parāśarād yogī
jātaḥ
was begotten
parāśarāt
by Parāśara
yogī
the great sage
vāsavyāṁ kalayā hareḥ
vāsavyām
in the womb of the daughter of Vasu
kalayā
in the plenary portion
hareḥ
of the Personality of Godhead.
Sūta Gosvāmī said: When the second millennium overlapped the third, the great sage [Vyāsadeva] was born to Parāśara in the womb of Satyavatī, the daughter of Vasu.
Purport

There is a chronological order of the four millenniums, namely Satya, Dvāpara, Tretā and Kali. But sometimes there is overlapping. During the regime of Vaivasvata Manu, there was an overlapping of the twenty-eighth round of the four millenniums, and the third millennium appeared prior to the second. In that particular millennium, Lord Śrī Kṛṣṇa also descends, and because of this there was some particular alteration. The mother of the great sage was Satyavatī, the daughter of the Vasu (fisherman), and the father was the great Parāśara Muni. That is the history of Vyāsadeva’s birth. Every millennium is divided into three periods, and each period is called a sandhyā. Vyāsadeva appeared in the third sandhyā of that particular age.

— A.C. Bhaktivedanta Swami Prabhupāda
Gauḍīya Ācāryas
Viśvanātha Cakravartī Ṭhākura Gauḍīya
Translation
Sūta said: Vyāsa was born in the womb of Satyavatī as a portion of the Lord when the third part of Dvāpara-yuga arrived in the passing of yugas.
Commentary

In answer to the questions mentioned in verse 3 “in which yuga, at what place” a short account of Vyāsa’s birth and activities is now presented. According to Amara-koṣa, paryaya means a lapse. With a lapse of many yuga cycles (yuga-paryaye), in Dvāpara-yuga, when Kṛṣṇa appeared, Vyāsa was born. It will be explained that Kṛṣṇa appeared in Dvāpara-yuga of the twenty-eighth cycle of yugas in Vaivasvata-manvantara. All yugas are divided into three parts: the beginning portion (saṇdhyā-rūpa), the middle portion (yuga-rūpa) and the end portion (sandhyāṁśa-rūpa). Tṛtīye refers to the third part of Dvāpara-yuga. [Note: The proportions are .1 for the beginning portion, .8 for the middle portion and .1 for the concluding portion of the yuga.] Vyāsa was born from Satyavatī who was the daughter of Uparicara Vasu. Thus vāsavyām means “in the womb of Satyavatī.”

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