Override parent theme function that is not hooked or in the functions.php file

Is this even possible?

If your description is correct and …

  1. The function is not pluggable, by which I assume you mean not
    wrapped in a if (!function_exists()) condition and
  2. The function is not a callback for an action or a filter and
  3. The function is not in a theme file that gets automatically substituted by the template loading system

… then you can’t remove it or replace it except by editing the parent theme file where the function is defined, and then, yes, unless you replace that function with an altered but compatible version with the same name you will get errors. And edits to the parent will be overwritten when the parent is updated.

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