Wordpress - Include files in child theme functions file

Child themes reference parent themes by directory name, and in a normal install all your themes live in wp-content/themes/, so I'd say it's fine to reference those themes by their relative path:

include '../parent-theme/some-file.php';

If that makes you uncomfortable, I observe the following constants in WordPress 3.0.1 with a twentyten child theme called tt-child:

TEMPLATEPATH     /home/adam/public_html/wp3/wp-content/themes/twentyten
STYLESHEETPATH   /home/adam/public_html/wp3/wp-content/themes/tt-child

So you can do the following in your child theme to reference the parent theme directory:

include TEMPLATEPATH . '/some-file.php';

In a child theme the proper way is

require_once( get_stylesheet_directory() . '/foo.php');

While in the parent theme you can still use

require_once ( get_template_directory() . '/foo.php' );

get_template_directory() still works in the child theme, sadly target the parent theme directory. In your case it's useful

You definitely do not want to hard code the URL. The proper way of doing so is

require_once( get_stylesheet_directory(). '/my_included_file.php' );

See more info at Wordpress Codex

Now, if your e.g. modifying header.php which has an include you would reference it as follows:

require_once( get_stylesheet_directory() . '/../parenthteme/my_included_file.php' );