I’m assuming he is not a minor and just happens to be living with his parents. I must agree with Dave G. If the home is that of another, respecting the desires of the homeowner is appropriate. If someone was visiting us, I would expect them to adhere to the rules of our home.
With respect to the initial question, using a safety deposit box creates unusual problems for purposes of storage.
While the bank may have its own policies and rules, we are concerned with what can legally be done without becoming subject to criminal penalties.
In some states, there are prohibitions, statutory, ordinance, or administrative in nature) from taking firearms into certain buildings. He needs to make sure that it legal to carry a firearm into a bank in his jurisdiction? In my state, for example, the storage of a firearm in a safety deposit box would be a clear violation of the law for most people (which specifies the only places where you can have a firearm), a conviction of which brings mandatory penalties.
I assume the firearm would need to be in some case or covered in some fashion. Assuming there are no problems with the location, at what point does the firearm become concealed under the laws of the jurisdiction?