Thanks for this interesting question - I've had a fair bit of a think about it - hope the answer's helpful...
If there are 6.5 billion people in the world currently alive, that does make you 6.5 billion
 times more likely (given that you are currently alive) to be someone else than you - at least from my
 perspective. Though, in fact, from _my_ perspective you are probably more likely to be you since
 not all of those 6.5 billion have internet access, and even fewer know about The Answer Shop!
_However_, since the universe is infinite in size and age, there were/are/will be an infinite
 number of living beings _you_ could have been (from an objective perspective), so your chances
 of being you (_or_ for that matter, _my_ chances of being _me_) - from an objective perspective - 
 are actually 0+ (by which I mean an infinitesimal amount).
_However_ again, the fact that _you_ _are_ you, (from your perspective) means that that is the _only_
 person you could possibly have been - and there is _no_ luck or probability involved - it's a
 dead cert! (i.e. probability 1)
To _mis-_quote a line from one of Pierce Brosnan's James Bond movies I saw recently - "everything
 that happens is luck (aleph0 = infinity) except destiny (aleph1 = zero)"
Please bear in mind that everything I say is necessarily from _my_ perspective!! :-)
As regards your parents, from an objective, mathematical, viewpoint, if one accepts that their
 probabilities of being themselves are (approx) 6.5 billion to one, then one could fairly argue
 that the probability of you being born to the pair of them is (in the face of no other evidence
 i.e. all other things being equal) approx (6.5 billion x 6.5 billion) to one. However, there
 are several reasons why such a view is oversimplistic, even mathematically, because, for instance
 I am sure neither of them chose each other from an equally likely pool of all living humans - 
 most of which they wouldn't even have met (for geographical reasons if no other). It is always
 important when calculating probabilities to state clearly (and often in some detail) your assumptions 
 as these are directly related to prior probabilities which (as Bayes made clear) can have
 dramatic effects on results of probabilistic calculations.
Your question is a very good example of the above:- given an objective standpoint your chances of
 being you are zero. Given _your_ standpoint (I'm assuming) your chances of being you are 1. You
 can't get much more differing probabilistic results than that!