How to override parent functions in child themes?

You should run the code after theme setup.

function osu_twentyten_continue_reading_link() {
    return ' <a href="'. get_permalink() . '">' . __( 'Read on <span class="meta-nav">&rarr;</span>', 'twentyten-child' ) . '</a>';
}

function osu_twentyten_auto_excerpt_more( $more ) {
    return ' &hellip;' . osu_twentyten_continue_reading_link();
}

function my_child_theme_setup() {
    remove_filter( 'excerpt_more', 'twentyten_auto_excerpt_more' );
    add_filter( 'excerpt_more', 'osu_twentyten_auto_excerpt_more' );
}

add_action( 'after_setup_theme', 'my_child_theme_setup' );