Save post_parent in a custom post type

You could use the wp_insert_post_data filter instead, which gives you direct access to the post’s data.

function wpa_insert_post( $data , $postarr ){ 
    //do something with $data['post_parent']
    return $data;
}
add_filter( 'wp_insert_post_data' , 'wpa_insert_post' , '99', 2 );

UPDATE:

I added the following to your code and it’s working:

function wpa_insert_post( $data , $postarr ){ 
    //do something with $data['post_parent']
    $data['post_parent'] = intval($postarr['post_parent']);

    //update meta data
    update_post_meta($postarr['ID'], 'author', $postarr['author']);
    update_post_meta($postarr['ID'], 'price', $postarr['price']);

    return $data;
}
add_filter( 'wp_insert_post_data' , 'wpa_insert_post' , '99', 2 );