Personally I don't think so. As /u/GaiusPompeius said Ovid was big into to transforming animal gods in the metamorphoses. There are other scattered references around to zoophilia, Herodutus mentions that Egyptian women engaged in it for mystical/religious purposes (Book 2) and this is echoed by Strabo among others. While there is no way to prove them wrong it is also possible that 1) they got misinformation 2) made it up to make the story seem more exotic.

More illuminating I think is that nowhere in Roman law is there a prohibition against it, while in the Levant there are extensive laws against it (Leviticus 20:15), commanding hat the guilty man be put to death and the animal killed. The Roman jurists never saw the need to legislate on the matter when they intervene in seemingly trivial matters like the suppression of Bacchic cults and the amount of babies per family. Either this means that they didnt care or that so few people were having sex with animals that they saw no need to.

We should also bear in mind that people like to fantasise about stuff. While there may be many depictions in art and literature that in no way means that everyone was fucking goats.

First, there's no evidence of bestiality playing a role in religious ritual. Myth is different, and even then, the gods posing as animals is different than actual bestiality. Additionally, many things depicted in myth were abhorred by Greeks and Romans, such as human sacrifice.

As for its acceptability, it's hard to say over such a long span of time as Greek and Roman culture has. However, the sources we do have are pretty firmly opposed to bestiality. Look, for example, at the myth of Pasiphaë. She was the wife of king Minos, and mother of the minotaur. I'm a Latinist, so I'm not as good with Greek sources off the top of my head, but here's a Roman source, Vergil:

hic crudelis amor tauri suppostaque furto

Pasiphae mixtumque genus prolesque biformis

Minotaurus inest, Veneris monimenta nefandae

(Aeneid VI.24-26)

Which translates to:

Here the cruel love of the bull [EDIT: just to clarify, I'd say this is an objective genitive, meaning basically "love for the bull"] and Pasiphaë placed under by deceit, and the mixed kind, and the two-formed offspring the Minotaur is on [the artwork being described], the reminders of unspeakable love.

Nefanda is a very important word here; it means "unspeakable," and to the Romans, it was reserved for some of the worst crimes.

So, seeing this reaction to the bestiality that begat the minotaur, we can understand how Romans of the 1st century BC, at least, would have reacted to bestiality.

Did Jupiter do any actual banging? Zeus did, and there's plenty of stories that tell of that. While Jupiter was equated with Zeus, I remember that Jupiter had a different role (and was "used" differently) than Zeus.

The only major deities in the Roman/Italian pantheon that I can think of who did any banging were Mars (and from what I've read, possibly had fertility rites originally attached to him), Priapus (that ol' 5th limb sure needs to do something), and Pan - goats, fauns, etc.

This is later than the Ancient Greek/Roman society, but...

Around 500CE the woman who would become Empress Theodora was an actress in a sleezy sideshow theater, and at that time this meant quasi-prostitute.

She was semi-famous for her depiction of the myth of Lydia and the Swan using a live swan on stage. Extant records don't say whether this meant she actually had sex with the swan on stage (unlikely I'd wager) but her performances were described as scandalous and salacious.

So there's at least one data point from 500CE showing bestiality as a titillating but taboo subject, at least somewhat similar to many attitudes today.