I’ll start by sharing my English treatment of Bhāgavatam’s 4.1.2 through 6 – a section that involves marriage between twin siblings and, if examined with care reveals the essence of Rādhā and Krishna’s parakīya-bhāva.
With his wife’s approval Manu allowed his daughter Ākūti (inspiration) to marry the sage Ruci (desire), on the condition that Manu and Śatarūpā would raise the couple’s boy as their own son.
Ruci – a blessed progenitor with great spiritual realization and exalted meditations – produced twins with Ākūti. They named the boy Yajña and the girl Dakṣiṇā, because inspirations (ākūti) and desires (ruci) are fulfilled when they lead to efforts (yajñā) that successfully invoke rewards (dakṣiṇā).
The boy was Viṣṇu himself, and the girl was an expansion of Goddess Bhū, Viṣṇu’s consort. Manu was delighted to raise the extremely brilliant boy in his own home. Ruci was delighted to raise the girl. After being raised separately, their eternal, unstoppable love brought them together in marriage, and they were delighted to produce twelve children.
This illustrates the essential principle of parakīya-bhāva. Parakīya is the most exalted form of love because it destroys all obstacles. In fact it enjoys expressing its unstoppable force by having obstacles to destroy. Parakīya is most famously manifest in Krishna-līlā. Krishna doesn’t marry the Gopīs. They are married to other men. Why? So that parakīya can express its power by destroying that obstacle to their love.
As is the case in this example, obstacles to love very often take the shape of social restrictions. For us, social restrictions have real utility, but for liberated beings, the only purpose of social restrictions is to facilitate their play. The ultimate reason for social restrictions is to facilitate the play of divinity. The byproduct of social restrictions is that they benefit conditioned beings. The concept of marriage originates in not in the need for stable child rearing, etc. It originates in the desire of the Supreme to express a love so powerful that it will destroy everything else, all other conventions, all restrictions. So marriage originates to facilitate the parakīya mood of Krishna and Rādhā. As a byproduct, marriage becomes the basis of svakīya-rasa, and as a more remote byproduct marriage generates social conventions that happen to be very helpful and useful in the lonesome world of cause and effect.
In this particular case, with Yajña and Dakṣiṇā, parakīya-rasa is not literally the sense of being unmarried. Yajña and Dakṣiṇā duly married with Brahmā, Manu, Śatarūpā, Ākūti, and Ruci’s blessings. In this case, however, the essence of parakīya manifests by breaking the social convention that siblings must not marry.
The purpose of this convention for ordinary souls is to keep family life sane and safe, and, even more practically, to keep human DNA healthy. Yajña and Dakṣiṇā are not human beings. In essence they are Viṣṇu and Bhū / Krishna and Rādhā, and even in appearance they are born amongst the supreme, super-human gods. Their mother was a supra-human because she was the immediate daughter of Manu, and their father was super-human because he was created directly by Brahmā. They lived at the dawn of universal time, when there were barely a few beings in the entire world capable of organic reproduction. For all these reasons, they had no practical need to observe the convention against marriage between siblings. Further, they were intentionally seperated at birth (due to the forethought of Śatarūpā) and were not raised as siblings, so their romantic mood never changed into the mood of being siblings. Thus they could take the opportunity to enjoy the extreme bliss of parakīya bhāva by expressing romantic love so strongly that all rules and conventions standing in its way were destroyed.
– Vraja Kishor