Override theme function in child

Hi,
I want to override a theme function handling an AJAX call, which that is NOT present in the functions.php file but in a file inside a directory.
Here it is:

Parent folder:
~parent_theme/functions/ajax.php – here I have a function called ‘parent_ajax_send_email‘
The function is attached to the event wp_ajax_nopriv_parent_send_email, built in the wp-admin/admin-ajax.php file, and run with the do_action( ‘wp_ajax_nopriv_’ . $_REQUEST[‘action’] ).

Child folder:
I have copied the parent file into the mirror location, i.e. child/functions/ajax.php file, changed the function name to ‘child_ajax_send_email‘ then added the following in the functions.php file:

// Replaces the parent theme ajax send email function
function replace_parent_ajax_send_email() {
	remove_action('wp_ajax_parent_send_email', 'parent_ajax_send_email');
	remove_action('wp_ajax_nopriv_parent_send_email', 'parent_ajax_send_email');

	//add_action('wp_ajax_child_send_email', 'child_ajax_send_email');
	//add_action('wp_ajax_nopriv_child_send_email', 'child_ajax_send_email');
}
// Call 'remove_parent_ajax_send_email' during WP initialization
add_action('init','replace_parent_ajax_send_email');

NOTE: I have also tried with this (with same result):
add_action('after_setup_theme','replace_parent_ajax_send_email');

Of course, I have changed in the JavaScript calling function the name of the action (which works because, I’ve checked, it goes to wp-admin/admin-ajax.php.

What happens is that the call goes with the right action request (modified to the child’s one) to wp-admin/admin-ajax.php, but the function do_action( ‘wp_ajax_nopriv_’ . $_REQUEST[‘action’] ) there returns ‘null‘, showing that the replacing function has not been registered with WP.

I believe there is a wrong new function registration with WP (because is not in the child’s root directory), but no idea how to solve it without touching the parent’s code.