Most of the younger collection of Olympians are most commonly said to have been sired by Zeus, with a few exceptions of gods who are presented with alternate fathering, this may have been as part of a development at some point in Hellas history in which many of the dominant gods were arranged as his children in order to justify their relationship with him and their belonging to his rule (and agreeability to it). That is to say that being fathered by Zeus may have rendered some sensibility to the governance of these deities with realms of function that are particular to Zeus’ spiritual aetheric rule. Apollon presented in Crete as being a son of Corybas, and one who competed with Zeus for rule according to local myth, is an excellent example of the son of an earthy almost chthonic god. His father being a great dragon of a god and follower of Rhea. Apollon here is one who, despite also being a earthy god, like his father, associated with herding and fruition of crops, was born possessing such similar powers of aetheric light and power that it makes him a natural rival of Zeus. A rivalry that, upon loosing to Zeus, was neutralized and Apollon was born as a son of Zeus through the ambiguous lady Leto. The parallel nature of Zeus and Apollon in many respects seems to play out in Orphic hymns in their common identifications with Pan and Helios. Apollon became a favored son of his father and one so, understandably, intrinsically a part of the father god that he often acted as an extension of his father’s will through carrying out his father’s designs and his role as oracle.
Yet two goddesses of alternate parentage are also apparent, daughters of Poseidon. One is Artemis, the daughter of Demeter and Poseidon, a tamer of horses, and holding certain fame as such throughout most of the Peloponesse. The other is Athena who was regarded as the daughter of Poseidon particularly in Crete and Libya, the yoker of horses. Edward Butler, in conversation with him, made a great case for the fathering of these goddesses by Poseidon firmly attaches them to the physical world, whereas the fathering by Zeus attaches them to the spiritual world of the intellect (my paraphrase..he is welcome to elaborate here in comments below!). As such the fathering of these goddesses serve very particular cases to the nature of these goddesses which is a bit different from the alternative parentage of Apollon shown above In the case of Athena we have a goddess who is looking to separate herself from her parentage by her own power. I think that this can be part of an understanding of her being conceived by Metis. She becomes a daughter of Zeus by her own volition, surrenders her power unto to him and in a way becomes a part of him as he mythically consumes her via the myth of consuming her mother and she emerges again from his head, her nature completely enjoined with his as his nous. It is not hard to imagine that her original form may have been very close to the aquatic dwelling feathered sirens as an archaic bird goddess with chthonic attachments (considering that the realm of Poseidon touches on both the heavenly and the chthonic it is easy to see why her is so fluid in her domain between these principles in more subtle ways…such as her association with the gorgon Medusa which may have been a reflection of an earlier identity of the goddess that the death of Medusa by her involvement becomes part of her harnessing and control of her earlier chthonic leanings under her more elevated form).
As such Athena provides an intermediary service, the earthy horse of Poseidon (which in some mythic version he produced out of competition with her for Athens, and in other mythic versions he produced as a gift for Demeter) becomes yoked by her and by her efforts transformed into a winged spirit horse, even as the death she orchestrated for Medusa yields the winged Pegasus. Even though she is no longer identified or a part of Poseidon her natal relationship with Poseidon is effective in her transformative and elevating principles. She weaves the cloth of the chrysalis which surrounds the earthy and works to transform it into the beautiful spiritual being. She is thus operating on the threshold of Poseidon’s domain into the aetherical.
Artemis is another story, however. She is firmly planted within the domain of the earthly and works from the opposite end of Athena. This can be best seen as her role and huntress/nurturer. She propels forward from the earthly state, driving forth all. She is the tamer of the horse, the one who gentles that which is wild and uncontained (energy) in order that it service life to move forward and evolve. When Odysseus’ horses ran off wild, he prayed to her and later erected a temple to her as a tamer of horses when he had recovered them by the shore. This may in some manner echo the way Hippolytus is portrayed dying in which he, by the shore, driving forth his horses, is recovered by Poseidon who sends his great bull to slay the youth by causing his horses to throw him from his chariot in their fright. As favor to Theseus, Poseidon undoes the art of Artemis. Other versions of the myth say that Asklepios revived him and the went to Italy to dwell there with his goddess. As such this could also be interpreted as a boon to his daughter Artemis.
Unlike with Athena, we see very little interaction between Artemis and her other father Zeus, other than her mythic birth as his daughter and her gifts of sovereignty given to her by him as he did for Apollon. Most of her domain is fully entrenched in the domain of Poseidon. This is just more evident in the Peloponnese. In fact Leto as a constant companion of Artemis would tell us more that this is more to reinforce her position as the daughter of Zeus, which was more successful in other parts of Hellas. Meanwhile in the Peloponnese’s Artemis retained her more gorgon-like features and Potnia theron winged form was familiar from Boeotia to Sparta to Olympia. We also see Artemis Eurynome as a river dwelling goddess in a mermaid like form like responsible for the after birth nurse care and purification of Zeus. From her iconic charms in Sparta depicting her face with horses heads and her widespread cult as the daughter of Demeter and Poseidon in this part of Hellas, it is easy to see how by her more earthy nature while still possessing heavenly illumination that she was confused with Hekate who enjoyed similar popularity in other parts of Hellas as they possessed very similar natures as goddesses of the ways, divine nurses, and goddesses of beasts, lamination, nocturnal activity and their associations with the earth, heavens and seas. The distinction being that as a titan daughter of Asteria and Perse Hekate already had all three jurisdictions, and Artemis came into her heavenly one later which like accounts for myths in which Hekate is said to have reared Artemis.
Unlike Athena, Artemis remains very much the daughter of Poseidon as much as that of Zeus, if not more. Her greatest acquisition perhaps from her fathering by Zeus is her role of illumination and her association with Apollon whom she acquired as her mythic twin, in all probability to their perfect parallel-sameness of natures. They were in all ways one and so in their adoption by Zeus they became his twins, his twin lights. In this manner you have Apollon (likely the son of Rhea/Samothrakian Demeter children the Korybantes served, and Artemis as daughter of Demeter) “reborn by Zeus through Persephone as under the guise of Leto which would hold with the multi-facet way that they would enter into the mysteries and how later philosophy would say that Apollon Zeus and Dionysos are one while Hera Artemis and Persephone are one. Likewise serving a very important parallel in the mysteries in which Artemis and Apollon are concerned with the leading of initiates, leading along the way on an earthy liminal level, whereas Hermes and Hekate perform a parallel role for the divine goddess. Differing echoes of related domains that is tied together by this mixed parentage.
In fact, as Dionysos was also called the son of Hades, it is also possible that Apollon as the son of Corybas was the son of Poseidon under a local cult epithet related to the cult of Rhea. Given the popularity of Poseidon in parts of archaic Hellas it would certainly stand to reason that he would send out his son to counter the rising sovereignity of Zeus (remember not all myths have Poseidon being swallowed, but also have him reared with lambs placing him in similar juxtaposition with herding that we find Apollon who tends to the pastures and herds). The triad of Zeus-Poseidon-Hades thus becomes overlapped with a triad of Zeus-Apollon-Dionysos to relate to similar concepts, but one under the supremacy of Zeus overall.
There is certainly a rather oppositional character between Zeus and Poseidon through many myths, with the exception of the myth of Troy, and this was only because Poseidon was pissed off. We do see an instance where Athena conspires with Poseidon (and other gods) to overthrow Zeus, but this is a rare testimony to her relationship with Poseidon. All that is really apparent from her parentage by him is her nature (like Apollon) as a charioteer, and yoker of horses. Apollon, meanwhile enjoys a much more stable and constantly overlapping relationship with Poseidon, not only as a charioteer and builder of walls, but also given his many associations with the seas. But only Artemis was revered continually as a daughter of Poseidon even as she was worshiped as a daughter of Zeus as firmly entrenched as she is in her love for the earthy realm with its lush woods, rivers and seas enacting as a nurse to all things.