Current Post Parent Category & Child Category Links

Display the categories (or terms from other taxonomies) assigned to a
post ordered by parent-child category relationship. Similar to the
function get_the_category_list() which orders the categories by name.
This example must be used inside the loop.

<?php
$taxonomy = 'category';

// Get the term IDs assigned to post.
$post_terms = wp_get_object_terms(
    $post->ID,
    $taxonomy,
    array(
        'fields' => 'ids'
    )
);

// Separator between links.
$separator=", ";

if ( ! empty( $post_terms ) && ! is_wp_error( $post_terms ) ) {

    $term_ids = implode( ',' , $post_terms );

    $terms = wp_list_categories( array(
        'title_li' => '',
        'style'    => 'none',
        'echo'     => false,
        'taxonomy' => $taxonomy,
        'include'  => $term_ids
    ) );

    $terms = rtrim( trim( str_replace( '<br />',  $separator, $terms ) ), $separator );

    // Display post categories.
    echo  $terms;
}

Source: user contributed example from the Developer’s Code Reference.