how to pull parent term from nested array?
If you check closely with the var_dump() output, you will see, that you actually load an array of entity objects with the type taxonomy term: https://api.drupal.org/api/drupal/core%21modules%21taxonomy%21src%21Entity%21Term.php/class/Term/8.5.x

This object has a getter method getName() that returns the desired property:

$names = [];

foreach($parent as $term) {
  $names[] = $term-&gt;getName();
}

$names_list = implode(', ', $names);