My current character is a Hermetic maga living in the Pyrenees. She has Goetic Summoning, and has taken the ghost of a Gifted Sorginak (a regional Folk Witch variant) as her familiar (after courting them for a while in play) as per the "Goetic Magic" section of RoP:I that states that spirits can be taken as familiars.
As a Gifted character, the familiar should be able to learn the Parma Magica. The question remains, however, whether teaching her would be illegal or not.
I'm divided on it, but I noticed none of the Hermetic ghosts in Calebais use Parma (although mine is first edition). So, I presume they can't.
That being said, I can see a ghost Criamon teaching it when reviving one of the dead Paths.
My dodge would be that the Gift is an attribute of the soul, not the spirit, and as such, ghosts can't use it, simulating it with Might instead.
Most ghosts, except perhaps some created through mysteries, won't have the gift anymore. That doesn't mean they don't remember what they were before, however, or that they have none of their powers.
Well, aren't familiars treated as part of a hermetic magus's magical power? Then, legally would it be any different than the Magus knowing the Parma?
"The ghost of a Gifted Sorginak" is different from "the Gifted ghost of a Sordinak".
If the ghost has the Gift indeed (nothing explicetly forbids it AFAIK, but it would be unnusual) he should be able to learn Parma. However, the Parma Magica is only taught to members of the Order, and a familiar isn't one. It has none of the obligations nor any of the rights of a magus. So I'd lean towards it being illegal.
Additionally, while nothing stops a magical creature from having the Gift, keep in mind tat it should set the character at magus level. Generally speaking, I don't think the Gift (or the Gifted magic quality, from RoP:M) is compatible with a character intended to be a familiar.
The same is the case for apprentices. They are also part of the magus' magical power.
... and the matter of apprentices and parma is debated in «Servants of God» in Thrice Told Tales.
Quite.
Note also, that (ArM5 p.105):
Familiars learn Abilities in the same way as humans. They cannot, however, learn magic, ...
So (with RoP:TI p.124) binding a verrry rare still Gifted spirit as a familiar would at best freeze that spirit's magic forever. Why would that spirit consent to it, if it still has the Gift enabling it to learn much more? Why would a maga admiring the spirit do that to it? Shouldn't she have tried to make it a covenant member instead?
AFAICS the SG got that Goetic maga painted into a corner with an extremely unlikely familiar, and needs to get her out there on their own. Reading rules on a forum does not help here. The SG just needs to understand the maga and her situation in depth and act accordingly, likely resorting to house rules.
Anyway, as a familiar even if Gifted cannot learn magic, it should not be able to learn the Parma Magica ritual either - or should it?
I wont weigh in on the discussion as to whether the spirit in question can use the parma magica in this post.
As to whether the familiar can acquire a score in Parma magica, I would say that the answer is unequivocally yes. I say this because the rules are quite clear that familiars can acquire xp in the same way that a magus can, which means that it is possible for the familiar to e.g. read books about parma magica or be taught by a teacher.
I am of the opinion that mundanes can also learn the parma magica in this way. They will derive no benefit from doing so since it is necessary to have the Gift in order to actually gain any benefit from performing the Parma magica ritual. They will be a huge liability for the order of hermes, because they can pass their knowledge of Parma magica on to other Gifted people.
There is a question here of how much "understanding the Parma Magica" is possible for a mundane person. I would, for instance, be wary about a mundane gaining a score in Finesse or Penetration, because it feels so tied in to a particular practice, whereas we know from canon that mundanes can learn Magic Theory.
The question of "which one is Parma more like?" seems open to me, but I lean towards the former - too tied into magical practice (which a non-Gifted person just cannot comprehend in the same way). I would maybe rule that a mundane who studied a book on the Parma gains xp in Magic Theory, with the option for a speciality in the Parma Magica.
The question isn't "Is this possible," it's "What kinds of legal arguments would this provoke if it did happen."
First of all, note that there is no specific provision in the Oath against teaching Parma. It's "illegal" only because it is widely accepted that leaking the secret of the parma magica to outsiders endangers the Order at large.
So, if the magus could make a point that teaching the Parma to his familiar presents no greater danger than, say, himself alone knowing the Parma, then there would be no legal ground against him - except perhaps in some specific Tribunal/epoch etc. Given how easy it is to summon and bind ghosts, however, I am afraid this argument would be relatively weak (of course, a weak argument by a politically strong magus can often carry the day).
Also, it should be noted that in order for the legal case to be brought up, someone ought to learn about the issue. This is not easy at all, as it would probably require scrying on the familiar, which is something that few magi would risk doing... and in a fairly specialized fashion too, distinguishing a parma originating from the familiar from one originating from the magus and "extended" in some fashion to the familiar.
Tangentially, I disagree with Oneshot when he interprets the text in ArM5 as saying that no familiar, even Gifted, can possibly learn any magic, period.
This is reading the passage entirely out of context. What it's intended, I believe, is that the familiar bond allows familiars to learn abilities "better" than they would normally be able to, but it still does not allow them to learn magic any better. However, it does not preclude a familiar who would otherwise be able to learn new magic from doing so. In this sense note that, while Timothy Ferguson's view, namely that:
may apply to the Gift, magical creatures can explicitly have a major magic quality (Gifted) that effectively gives them "something very similar" to it and allows them to learn supernatural abilities etc.
While the Oath may not say anything about the Parma Magica, I am quite sure the Peripheral Code has something to say on the matter, and that is just as legally binding on magi.
I would think that a familiar could only be taught parma after swearing the oath of Hermes.
At which point you would have other issues along the lines of whether they were a member of the order and whether they could reman as a familiar as a member- I expect this would eventually be ruled in favor of the familiar being retained since it is a voluntary relationship, but it could take some time and drama (aka stories) to get resolved.
Exactly. Parma not being explicitly mentioned does not make it any more legal; endangering the Order is illegal no matter what danger or how. There is a second reason why teaching Parma causes a problem; it arguably invokes the «join or die» clause. Anyone learning Parma is evidently powerful enough to matter and hence must join or die. This is the reverse argument of @silveroak 's. If you do not have to swear the Oath before you learn Parma, you have to swear it afterwards.
It is a can of worms.
Incidentally in our saga,
we have played two thirds of «Servants of Gods», with the players taking an unanticipated approach. Learning that the apprentice knew Parma Magica, the resident quaesitoris gave him a 24h crash course in the Code and swore him into the Order. Thus she arguably break the Code by depriving the master of his magical power, but that argument remains to be resolved.
We had a similar situation in my Thebes saga, where a Tytalus had kept an apprentice for over 20 years claiming he hadn't passed his gauntlet yet. Was a fun Tribunal debate, that one.
Honestly this makes me wish that the Opening of the Arts in the Hermetic Arts (which is something all members of the Order have to go through) is what gives people the key to the Parma Magica. Imagine if its the first thing learned, right after being initiated into the magic rather than the last thing.
So much better!
Also its less worrisome about someone talking about it or a Parma Book being stolen.
Until a just-gauntleted apprentice loses his parens to twilight and gets picked up by a member of another tradition before he's found by the Order.
Actually, the Order has members (in particular ex-Miscellanea ones) who are only nominally Hermetic - they do not know the Hermetic Arts. Also, exotic wizards joining the Order may not be able to have their Arts Opened (this was true of several, perhaps most, Founders too), being already too experienced in their non-hermetic magics. For setting consistency, it's crucial that they should still be able to learn the Parma Magica.
How dare you doubt Bonisagus' InVi lab total? Heretic! 
Well he did have a distinct lack of tractatus to study from.