I think this is something you must ask your A-kasse about.
I am also not aware of any form that needed to be filled in. Your A-kasse will just ask your home country's public administration responsible for registering residence for people in your country.
I am sure if you have lived there, it will also be registered, regardless of you have been living with your parents.
About other kind of proof, you must ask your A-kasse what they want. But I would assume that if you have been student, then it must be possible to prove that you was signed up for the School and get somebody to confirm that you actually have been present in the school. If you had a job while you were in your home-country you must have some kind of job-contract.
Be aware that this residence requirement is a new thing (have been in force only since January 1, 2019), so at the moment we do not have sufficient knowledge about how the requirement of proof will be interpreted by the A-kasser, i.e. what kind or how much they will ask for.
But all normal things that proof you have been living in a country, can probably be used - for example tenant-contracts, job-contract etc.
You must go into a dialogue with your A-kasse, and ask what they want of proof..