Can I stop the subscriber data in a child BU from being written to the parent BU all subscribers list? I understand that I can use Business Unit Filters to control which subscribers are visible in certain business units but, in order to adhere to organizational data compliance practices, I need stop the data contained in my child bu from being shared with the parent BU.
You can't stop subscribers from being added to the All Subscribers list. If a subscriber isn't in that list then they aren't a subscriber and you can't communicate with them.
You mentioned business unit filters but I believe that was an older feature that is only available on older accounts so might not be an option for everyone.
In a similar question, it was pointed out that access to the All Subscriber list is commonly blocked. This is the approach I have previously taken when data compliance practices need to be adhered to. This can be done via the user permissions.