Citizenship of a country by descent is normally automatic when the person is the child of another citizen of that country. It's quite common for people to hold citizenship of several countries, sometimes without even knowing it.
You state that you've looked at the PICA website but have you tried emailing the relevant department(s)? If you simply want to check whether you're already a Jamaican citizen (or to see if you're eligible to apply for citizenship), email
[email protected]
If you want to apply for a Jamaican passport, email
[email protected]
Chris