Hi @Andrey_Malk
As you are over 18 and have your own individual asylum case, then you can withdraw it yourself.
It is not the Ausländerbehörde that decides on an asylum application but BAMF. I cannot say whether you withdrawing your case will have an influence on your parent’s application. You would need much more in-depth advice on this but basically you have to consider whether any aspect of their fear of persecution in your home country is related to you.
I advise you to have a **** at the REAG/GARP programmes on offer by Germany. These provide financial assistance to people who withdraw their asylum applications and wish to return to their home country. It can help with travel costs but also a one-off payment which you could use to finance German courses. Here is the information in English (it is also available in other languages):
https://***.returningfromgermany.de/en/programmes/reag-garp/
You have to first contact an advisory service if you wish to apply for this. There are also additional programmes in specific countries. Here you can see if there are any such programmes in your home country:
https://***.returningfromgermany.de/en/countries/
A re-entry ban (Einreisesperre) only applies if a person has been deported from Germany. If a person voluntarily leaves Germany, this does not apply.
Best,
Éanna