If you are applying for asylum in the United States, and have no other basis for staying in the country legally, then you must wait until your asylum case is approved, plus one year after that, before applying for a green card (adjusting your status to U.S. lawful permanent resident) on that basis. If, however, you have some separate basis upon which to apply for a U.S. green card, parallel applications are possible.
For some people applying for asylum, the application process moves fairly quickly (unless you are denied asylum in immigration court and are appealing your case). Others, however, get stuck in bureaucratic backups from the beginning.
The agency in charge of this process, U.S. Citizenship and Immigration Services (USCIS) is notoriously backlogged. It's difficult to predict how long this step will take. What's more, applicants who aren't granted asylum by USCIS in most cases must proceed to Immigration Court for removal proceedings, where a judge might approve them for asylum (or order deportation).
If and when USCIS approves you for asylum, you will be able to apply for a green card one year later, through a process known as "adjustment of status."
No notification is sent to asylees to remind them to apply for a green card. You should keep track of the one-year anniversary of your asylum award, and apply for the green card (adjustment of status) as soon as you're eligible. That's because, if conditions in your home country improve and you no longer fear persecution there, your asylum status could be taken away. In fact, if and when you apply for a green card as an asylee, you might be asked to prove your continued fear of persecution.
If the processing of your asylum case has been delayed or is perhaps continuing on into appeals to higher courts, then it might make sense for you to pursue some alternate path to a U.S. green card. For example, you might marry a U.S. citizen, win a place in the diversity visa lottery, or find a U.S. employer willing to sponsor you.
Again, there is no prohibition on having more than one type of application pending with U.S. Citizenship and Immigration Services (USCIS) at the same time.
While a grant of asylum will offer you a long-term right to stay in the United States, it is not as secure as having a green card. Your right to asylum is, as mentioned, conditioned on your continued fear of returning to your home country. Once you have a green card, however, this condition no longer applies, and you can start counting the time until you are eligible to apply for naturalized U.S. citizenship, which is the most secure status of all.
Immigration law is highly complex, and submitting a winning asylum case requires detailed presentation of facts and legal arguments. If your case seems to be tied up in processing, perhaps it's because you need some help in making a convincing case for asylum.
Consult an experienced immigration attorney for help with both the asylum application and any other applications for a green card that you might submit at the same time.