Changing a parent theme safely

In short: Not easily.

First – just to say it: with child themes you can hook into functions, filter values, change templates or even exchange javascript files. This is usually enough.

Edit based on the comments: If the theme is coded with best practices and checks for child theme directory you may just copy the template from the parent theme to the child theme and do your changes there.

If child theme is not enough you have two possibilities:

  1. Write down your changes manually or backup the files. Once you update the theme reproduce the steps.

  2. Connect to your server with ssh. Install a git in the themes folder (or root if you need to). Check in your changes. Whenever you update the theme git merge the changes into the updated theme. Git is nothing for beginners. Make sure to backup your complete installation before experimenting with git.

There’s no other solution imho.