my $prepared_handle = $dbh->prepare('SELECT * FROM table WHERE parent_id = ?');

:
:
:
get_children(0);

sub get_children {
  my $parent_id = shift;

  $executed_handle = $prepared_handle->execute($parent_id);
  while (my $pointer = $executed_handle->fetchrow_hashref) {
    $results .= $pointer->{id}."\n";
    get_children($pointer->{parent_id});
  }
}