Handle all the class loading in your parent theme on a predictable action (point in time) and hook in later in your child theme.

Example:

add_action( 'wp_loaded', 'parent_prefix_load_classes', 10 );

function parent_prefix_load_classes()
{
    $classes = array ( 'Extra_Comment_Walker', 'Extra_Nav_Menu_Walker );

    foreach ( $classes as $class )
    {
        locate_template( "php/class.$class.php", TRUE, TRUE );
    }
}

Create instances in your child theme make sure your code runs after the classes are loaded:

// Priority 11 to run after the parent theme's loader.
add_action( 'wp_loaded', 'child_prefix_create_objects', 11 );

function child_prefix_create_objects()
{
    $nav_walker = new Extra_Nav_Menu_Walker;
}

Rules of thumb:

• Never load anything just when the file (function.php) is called. Wait for wp_loaded.
• Use the priority argument to control the order of execution.

Some notes:

• Custom post types, taxonomies and shortcodes belong to plugins. They should never be part of a theme, because that would create a lock-in effect for the user. If a theme switch would break the content you did something wrong.
• Do not use require_once in a theme. locate_template() is more flexible. You can overwrite the whole class now in your in child theme if you use the same directory structure.

Handle all the class loading in your parent theme on a predictable action (point in time) and hook in later in your child theme.

Example:

add_action( 'wp_loaded', 'parent_prefix_load_classes', 10 );

function parent_prefix_load_classes()
{
    $classes = [ 'Extra_Comment_Walker', 'Extra_Nav_Menu_Walker' ];

    foreach ( $classes as $class )
    {
        locate_template( "php/class.$class.php", TRUE, TRUE );
    }
}

Create instances in your child theme make sure your code runs after the classes are loaded:

// Priority 11 to run after the parent theme's loader.
add_action( 'wp_loaded', 'child_prefix_create_objects', 11 );

function child_prefix_create_objects()
{
    $nav_walker = new Extra_Nav_Menu_Walker;
}

Rules of thumb:

• Never load anything just when the file (function.php) is called. Wait for wp_loaded.
• Use the priority argument to control the order of execution.

Some notes:

• Custom post types, taxonomies and shortcodes belong to plugins. They should never be part of a theme, because that would create a lock-in effect for the user. If a theme switch would break the content you did something wrong.
• Do not use require_once in a theme. locate_template() is more flexible. You can overwrite the whole class now in your in child theme if you use the same directory structure.