NetWallah already told you what was the reason of the error. I would like to add a piece of advice:

Do not go over the elements in a loop (unless you want to add a student to almost each of them). Use XPath to search for the desired target:

#!/usr/bin/perl
use warnings;
use strict;

use XML::LibXML;

my $doc    = 'XML::LibXML'->load_xml( string => << '__XML__');
<?xml version="1.0" encoding="utf-8" ?>
<University>
    <students>
        <student id="1000"/>
        <student id="1001"/>
        <student id="1002"/>
    </students>
</University>
__XML__

my $query  = '//student[@id="1001"]';

for my $ele ($doc->findnodes($query)){
    my $new_ele = $doc->createElement('student');
    $new_ele->setAttribute('id', '1003');
    $ele->parentNode->insertAfter($new_ele, $ele) or die;
    last;            # Not needed if identifiers are unique. The whole
+ looping is useless, then.
}

print $doc->toString;
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