[plugin development]How to modify post content after post is saved?

Hello!

I’m developing WP plugin that needs to modify post content AFTER post is saved (so I’m sure it has and ID and permalink). I want it to fire only when users presses “publish” or “update” button (not for autosave).
My idea was to hook to “save post” action, change what I need and use $wpdb->update() to modify row in wp_post table. I came up with something similar to following code:

function myplugin_parse_post( $postId )
{
    global $post, $wpdb;

    $post->post_content = $post->post_content . '<div>my addition</div>';

    $wpdb->update( $wpdb->posts, array( 'post_content' => $post->post_content ), array( 'ID' => $postId ) );
}
add_action( 'save_post', 'myplugin_parse_post' );

After trying out my plugin I realized that this doesn’t work. I checked revisions and saw that they were modified, however changes weren’t visible in editor.
I tried a few different approaches (like using wp_is_post_revision()) but none of them worked.
Can you help me with this? WP plugin development is new topic for me and it is a little difficult to understand how exactly saving post works.
Thanks!