"How are your parents?"
The "rule" is that ご is mostly used with words that use Onyomi (Chinese) reading, while お is mostly used with words that use Kunyomi (Japanese) reading.
Though right in this sentence we have one of the exceptions 元気 (genki) which uses the Onyomi reading but still uses お.
Most words that consist of two or more Kanji use Onyomi readings but there are many exceptions to this rule as well.
(edit: you can also have a look at the link posted by trishka9 further down, that I hadn't seen when taking the time of writing this comment)
I saw in another thread that お is used before words of Japanese origin, while ご is used before words borrowed from Chinese.
This article http://selftaughtjapanese.com/2014/03/21/japanese-honorific-prefixes-%E3%81%8A-and-%E3%81%94-o-and-go/ explains it this way (and then adds some exceptions):
お : used for words with the 訓読み（”kunyomi”), or Japanese reading. It is sometimes written in Kanij as 御.
ご : used for words with the 音読み (“onyomi”), or Chinese reading.
'anata no' means your or yours or whatever comes after the 'anata no ' belongs to you, but because the question is asked to a spicific someone it is obvious that you are asking about his or her parents so you do not need to use anata no..
As for the 'go' particle, its job is to make your sentense more formal or just to show respect to the parents and it is not always necessary so it is correct to leave it
That is what I know about them : D
Yes, the ご. It's an honorific prefix for りょうしん. You wouldn't use it when talking about your own parents because using honorific prefixes for your own family/social circle when speaking to others is not humble and thus a no-no. It's the same reason you should refer to your father as 父 (ちち) instead of as お父さん (おとうさん) when talking about him to other people.
it should be お元気, not あ元気
It's also just a bit redundant to use あなたの in this case since the honorifics already imply you're talking about the listener's parents rather than your own. It's a bit rude to use あなた in general unless you don't know the listener's name at all and it can't be understood through context that you're talking about them. (Or if you are VERY familiar with the listener in which case あなた is similar to the pet name 'dear' or 'darling' in English)
Because that would be like asking "Is it the vitality of your parents?".
の indicates a possessive, which sort of glues two nouns together, and using it here would make 元気 （げんき） the subject/topic of the question. In the normal お元気ですか ("are you well?") 'genki' isn't used as a true noun.
It's certainly strange that it accepted that,
お and ご are both readings of the same kanji 御 (o for kun-yomi words, go for on-yomi), but the hiragana shouldn't be interchangeable in this case...
removing the honorific from 元気 isn't wrong either, though it is a bit rude when referring to someone else's parents.