You know, this is a really interesting question, with a few angles going on.
In general, the Greeks don't care if your animal parts look great, they still find them gauche. One of the major features of Greek aesthetic sensibility is the idea of absolute perfection of the human form, of which the gods are the most excellent example; no matter how bangin' awesome you look with your animal parts, they would always believe you'd look better without them. You're supposed to look like the pinnacle of the humanoid form, so even attractive animal parts still represent a deviation from perfection. An Ultimate Appearance god with a tail is always going to be considered doing it wrong, even if he's technically exactly as hot as an Ultimate Appearance god without one.
However, wings are specifically an exception in Greek mythology. Despite the general disdain for people who look inhuman (which, by the way, doesn't only include animal parts - dudes like Priapus are not winning any points, either), many gods do appear with wings on their shoulders without any apparent stigma.
Iris, the rainbow-goddess and messenger of the gods
Eros, god of love
Nike, goddess of victory in battle and competition
Various lesser gods are specifically said to be winged, and ancient writers seem perfectly happy to mention this without seeming to put any negative connotation on it.
The difference here between wings and other animal parts lies largely in their specific symbolism in Greek mythology. Where other animal parts usually represent bestiality, wildness and a lack of civilization and humanity, bird-like wings instead often represent swiftness and the ability to be ever-present, making them obvious choices for gods whose job it is to be everywhere all the time or to rush to and fro on their divine errands. Gods that travel constantly, like Iris, have wings to illustrate their fleetness, while gods that are capable of being present wherever needed, like Eros and Nike as the bringers of love and victory, have wings to show that they can instantly travel to anyone's side. Gods of swiftly-moving natural phenomena also often appear with wings, such as the gods of the four winds and their offspring, or Hypnos and Thanatos, lords of sleep and death, who must travel the world unceasingly to bestow their gifts on mankind.
Basically, in Greek iconography, wings are usually meant to suggest that the god that possesses them has Psychopomp or otherwise travels a lot, rather than meaning they necessarily have anything to do with birds or Animal. In fact, gods that do have to do with birds, like Aphrodite with swans, Hera with peacocks or Apollo with ravens, usually don't appear with bird parts in spite of their association.
So having wings might, depending on your Storyteller's interpretation, be the one permanent animal feature that a young god of the Theoi can get away with without courting social repercussions (the dreaded Greek God Snub). It really depends on whether your ST believes that those gods literally have wings - which they very well might - or prefers the interpretation that those wings are an affectation of Greek artists rather than a literal truth, in which case having actual wings might still be a problem for you. It's also worth noting that only lesser-Legend Greek gods usually rock the wings, which is most likely because grunt-work jobs like carrying messages or presiding tirelessly over human affairs are somewhat beneath the greater gods of Olympus, who are too important to be worrying about someone else's problems all the time. If you do choose to have permanent wings, it's likely that the other Theoi will automatically assume you're jockeying for one of these kinds of jobs, and to assign you to them even if that's not what you wanted; and if you do manage to get away from all of that, they're probably going to start questioning why you're being weird and wearing wings for no apparent reason.
Of course, you can always have animal parts regardless of what the Theoi think about it - it's not illegal to be creature-like, just considered in poor taste. As a new Scion trying to carve out a place for yourself, you can always decide to pal around with the other rustic gods and decide to ignore those stuffy old rules the geezers up on the mountain care about so much, or try to become a beloved figure among the Greek gods and start changing the status quo to become more accepting of alternative styles of beauty. But if you're looking to have a permanent animal feature (with Bestial Nature or relics or whatever else) and want to avoid the whole messy Mean Girls-esque political situation, wings are your best bet.
