Sometimes there are employment-based green card paths that can help for those seeking asylum but whether going down this path would allow you to travel would depend in part on your immigration history - what your status was when you came to the U.S., when you/your parents applied for asylum, etc. If you haven't already, you should speak with an immigration attorney to look at all options, including those based on employment.
That would be an employment visa, not green card. It can lead to a green card in some cases but it's complicated (depending on the country of origin it can take an extremely long time)
