Trying to get property of non-object when using $post->post_parent

@kaiser answer gives you the solution. I see from your comment that you don’t understand, so I upvoted his answer and translate it in code:

<?php
$bannerimg = 'home.jpg';
$post = is_singular() ? get_queried_object() : false;
if ( ! empty($post) && is_a($post, 'WP_Post') ) {
  if ( 'about' == $post->post_name || '2' == $post->post_parent ) {    
    $bannerimg = 'about.jpg';
  } elseif ( 'learning' == $post->post_name || '56' == $post->post_parent) ) {  
    $bannerimg = 'teaching.jpg';
  } elseif ( 'admissions' == $post->post_name || '15' == $post->post_parent ) { 
    $bannerimg = 'admissions.jpg';
  }
}