My father is a funeral director and his files don't ordinarily contain that sort of information, they only have specifics to do with the funeral: date, time, place, coffin details etc
The funeral director might have been privy to other 'family' information. If so that will have gone to his grave with him.
You could contact the crematorium and ask for more details about the funeral from their files. But I think that the only extra information would be the name of the applicant (next of kin or executor) and their address and how much it cost.