The US immigration laws specify that an immigrant – an asylum seeker – has the right to formally request to move their family to the US by filing Form I-730, Refugee/Asylee Relative Petition. If you obtained asylum through affirmative or defensive process, your spouse or a child below 21 years of age can be granted asylum as your derivatives, and you need to file a petition for each member of your family individually. That is, your direct relatives can derive asylum from you.
To request for asylum, your family need to provide documentation proving their relation to you.
Note that your parents, sisters, brothers, or cousins are not considered your direct family or, in legal terms, the first tier relatives. Also note that you must file Form I-730 within the two years that you received asylum in the US, unless there are solid proofs for the delay such as humanitarian reasons.