Are people born to non-Muslim parents unlucky?

No, there’s no injustice for them. They are not responsible for most of our duties. Things they are responsible for are: not associating something with Allah (accepting something else as a deity, an authority besides Allah), doing what is accepted as good by human nature and restrain themselves from what is accepted as bad by human nature. They are not responsible for the things which can be learned from a prophet. If they are informed by a prophet’s message, then they will be responsible to obey that prophet through his message.

See the article below for further information:

http://www.islamandquran.org/fatwas/what-is-the-responsibility-of-a-person-who-is-not-informed-of-the-quran.html