This is from BPHS (Bṛhat Parāśara Hora Śāstra)’s chapter, “Narration of Avataras,” ślokas 5-10
The basic idea is given at the end of śloka 8: सूर्यादिभ्यो ग्रहेभ्यश्च परमात्मांशनिः सृताः (sūryādibhyo grahebhyaś ca paramātmāṁśaniḥ sṛtāḥ). Incarnations appear in the world through the Paramātmā, which is also in the planets like the Sun and so on.
That’s the basic idea the book presents. It is similar to the mainstream, authoritative Vedānta Darśan which states that avataras appear in the word through Paramātmā (for example ŚB 1.3.5), with the addition that the author here claims that since the planets also are manifestations from Paramātma, we can extrapolate that the incarnations appear through the Paramātmā that is manifest in various planets.
Then he lists (ślokas 5 – the first half of 8) which avatāras appear via the paramātmā invested in which planets.
- Rāma from Sūrya
- Yādhunātha (Krishna) from Candra
- Nṛsiṁha from Bhūmi-putra (Mars, son of the earth)
- Buddha from Soma-suta (Mercury, son of the moon)
- Vāmana from Vibudhejyas (Jupiter)
- Bhārgava (Paraśurāma) from Bhārgava (Venus)
- Kūrma from Bhāskara-putra (Saturn, son of the sun)
- Sūkara (Varāha) from Saiṁhikeya (Rahu, son of Simhika)
- Mīna (Matsya) from Ketu
Then he says at the end of the first line of śloka 7 चान्ये तेऽपि खेटजाः (ca anye te ‘pi kheṭajāḥ). All the other incarnations also appear in this way (i.e. through the Paramātma – and therefore according to this author, in connection with the planets). This is authorized by ŚB 1.3.5 & 26.
In the remainer of śloka 7 and first half of 8 he makes another well accepted statement: Some incarnations (among these infinite many incarnations) are Paramātmā directly appearing in the world, others are Paramātma appearing through a jīva. We are familiar with the second type of avatār as “śakty āveśa avatāra.”
He further stipulates it in ślokas 9 and 10: When Paramātmā directly appears, we have avatāra like Rāma, Krishna, and so on. When Paramātmā empowers jīvas we get the species of living beings like humans, etc. This is authorized by ŚB 1.3.27.
The author of BPHS makes one more claim that may not be accepted in more important Sanskrit works, as far as I know. He says that when the avatara’s disappear from the world, they do so by the Paramātmā absorbing them back within himself. And, this happens through the same planetary Paramātmā agent as facilitated the manifestation of the avatāra.
Similarly he says the species of life merge back into Paramātma through the respective planets at the time of dissolution. And then everything merges back into the subtle substance (avyakta).
Thus, it is evident that the Author of this section has borrowed a widely accepted paradigm from the Vedānta Darśan and has inserted the planets into it as a way to invest astrology with more importance than it would otherwise have for the theistic Indian audience. The presence of planets in this scheme is not found, to my knowledge, in any other Sanksrit work – almost all of which are significantly more authoritative than the BPHS in terms of the topic at hand.