How do you apply for a birth certificate for a child born in the US by a Jamaican parent?

If the birth was registered, then getting a birth certificate is a straightforward process. The "Where to write for vital records" page on the CDC web site provides the contact information for each state. You can order a certificate in person or by mail.
Birth registration is compulsory in the United States, so if a child was born in a hospital, the birth was registered. The same is true if delivered by a physician or certified nurse midwife.

If the child was born at home without professional attendance, then it is the parent who is required to register the birth. The same contact links at the CDC can get you to a website or phone number where you can obtain the forms to register the birth.

Different jurisdictions have different laws and regulations, but in all the nationality of the parent doesn't matter.