You can try to inform the state parliament ("Landestag") or the "Bundestag" (the German federal parliament) of your situation by sending them a petition, in which you ask for the reconsideration of your asylum application or the right of residence in Germany. Keep in mind that you cannot send a petition and a hardship application ("Härtenfallantrag") at the same time. In principle, a petition has little chance of success. You must have strong reasons for the continuation of your stay in Germany; reasons which are not commonly shared by other rejected asylum seekers. You can send the petition as a normal letter to the "Landtag" or "Bundestag". However, it makes sense to try and personally contact individual members of the Landtag or Bundestag.
Please note: A petition has no suspensive effect on your possible deportation, i.e. you may be deported while the state parliament or the federal parliament is looking into your case.