Show only one page link in content pagination

To show prev/next links rather than numbered pagination you need to set the next_or_number argument to next.

Then if you want to remove the previous links, you can filter the HTML for the links with the wp_link_pages filter.

But, you can be craftier and filter on the wp_link_pages_link hook.

Although it is used in a number of contexts, when it is called for the previous link it passes a value representing the page number of the previous page. If you check this against the global $pages, you can come up with the right logic to decide whether you are filtering a “previous” link rather than “next” or numbered pagination.

add_filter( 'wp_link_pages_link', 'wpse_235716_kill_previous', 10, 2);

function wpse_235716_kill_previous( $link, $prev ) {

    global $page;

    if ( $prev > 0 && $prev > $page ) {
        return false;
    } else {
        return $link;
    }

}