display posts by category wordpress on a page code example

Example 1: wordpress display post categories

<?php the_terms( $post->ID, 'topics', 'Topics: ', ', ', ' ' ); ?>

Example 2: wordpress display posts from specific category on page

The argument isn't category, it is cat. Your query fails because you are using an argument that doesn't exist.

$args = array(
  'post_type' => 'post' ,
  'orderby' => 'date' ,
  'order' => 'DESC' ,
  'posts_per_page' => 6,
  'cat'         => '3',
  'paged' => get_query_var('paged'),
  'post_parent' => $parent
); 
$q = new WP_Query($args);
if ( $q->have_posts() ) { 
  while ( $q->have_posts() ) {
    $q->the_post();
    // your loop
  }
}
Notice that I have converted your query_posts() into a new WP_Query object. Do not use query_posts(), ever. Even the Codex states so.

Note: This function isn't meant to be used by plugins or themes. As explained later, there are better, more performant options to alter the main query. query_posts() is overly simplistic and problematic way to modify main query of a page by replacing it with new instance of the query. It is inefficient (re-runs SQL queries) and will outright fail in some circumstances (especially often when dealing with posts pagination). Any modern WP code should use more reliable methods, like making use of pre_get_posts hook, for this purpose.

http://codex.wordpress.org/Function_Reference/query_posts

Also note that I removed unnecessary PHP opening and closing tags and formatted the code for better readability. That alternative control structure syntax is a formula for failure, in my experience.