This is what you need
You need the following documentation:
Your Passport
Your Birth Certificate
Mother Passport - Copy
Father Passport - Copy
Mother and Father Marriage Certificate
(If this is from the UK it must be Apostiled, sent to UK Government to be verified and returned) This takes up to 30 days. If it's a Cypriot Marriage, not necessary.
If your parents were not married, they need to stand in court and swear an affidavit each that they are your parents and be court stamped. This is done in a Cyprus court or Cyprus Embassy.
Once you have all the above, you can go to the passport office in Cyprus and submit all docs. You will then wait 3-4 months to be given a Cypriot ID number. Once you have that then you can apply for your passport either in Cyprus or Embassy. Which will take a few weeks to get back.
