Just happened to be dealing with a situation somewhat similar to this one (I think).
In short, if – and only if! – the parent theme is well-written, with functions enclosed by if ( ! function_exists(‘function_name’) ) conditionals, then all you will need to do is place your replacement function in your child functions.php. In other words, there will be no need to re-produce the file structure of the parent theme. If you do need to include additional files other than template files, with or without the parent theme file structure, you’ll have to include or require them.
If, however, the function you wish to replace has not been conditionally enclosed, then you may have to look into removing it or its output via action or action hook. In some cases that might be sufficient. In others it might be difficult to figure out.
Can’t get more specific without seeing the rest of the code you’re dealing with!