After a lot of head scratching and cursing I noticed that the OPs data had a variety of quotes around the attribute values. I changed them to ordinary quotes and it now works ok.

kaif++ for spotting the snag.

#!/usr/bin/perl

use strict;
use warnings;
use HTML::TokeParser::Simple;
use Data::Dumper;

my $p = HTML::TokeParser::Simple->new(*DATA)
  or die "couldn't parse DATA: $!\n";
  
my (@records, %record, $start, $i);

while (my $t = $p->get_token){
  
  if ($t->is_start_tag('span')){
    if ($t->get_attr('class') and $t->get_attr('class') eq 'jobname'){
      $record{jobname} = $p->get_trimmed_text('/span');
    }
    elsif ($t->get_attr('class') and $t->get_attr('class') eq 'jobseri
+al'){
      $record{jobserial} = $p->get_trimmed_text('/span');
    }
    
    elsif ($t->get_attr('name') and $t->get_attr('name') eq 'em'){
      push @{$record{em}}, $p->get_trimmed_text('/span');
    }
    elsif ($t->get_attr('name') and $t->get_attr('name') eq 'offices')
+{
      $record{offices} = $p->get_trimmed_text('/span');
    }
  }

  if ($t->is_start_tag('blockquote')){
      next if $i;
    my $txt = $p->get_trimmed_text(('blockquote'));
    $record{job_desc} = $txt;
    push @records, {%record};
    %record = ();
    $i++;
  }
    
}

print Dumper \@records;

__DATA__
<p><b>
<span class="jobname">Accounting Assistant, Level 2</span>
<span class="jobserial">(19203)</span>
<br />Current members:<br />
<span name="em">Plow, Elliot</span> 
<span name="em">Wang, Susan</span>
<br />
<span name="offices">Huston</span>
</p>
<blockquote>
Job descriptions here. 
This block quoted text contains a job description 
and it what I am really looking to recover.
</blockquote>
<blockquote>
    <a href="#top">Go to the top of this page</a>.
</blockquote>
<blockquote>
    <a href="companyHR.html">Check for open positions now!</a>
</blockquote>
[download]

---------- Capture Output ----------
> "c:\perl\bin\perl.exe" _new.pl

$VAR1 = [
  {
    'em' => [
      'Plow, Elliot',
      'Wang, Susan'
    ],
    'job_desc' => 'Job descriptions here. This block quoted text conta
+ins a job description and it what I am really looking to recover.',
    'offices' => 'Huston',
    'jobserial' => '(19203)',
    'jobname' => 'Accounting Assistant, Level 2'
  }
];

> Terminated with exit code 0.
[download]