The approach is a two step approach - you get callbacks for three events, the start of a tag (start_h), the end of a tag (end_h) and anothe callback for any text encountered (text_h). So you need to set up a text handler that will see all text, and modify your start handler such that it increases a counter whenever it enters a <A ... tag, and your end handler such that it decreases that counter.

Your text handler then knows whenever it encounters text from within an anchor.

Some untested code that should replicate what I discussed :

# start of ParseLink
  {
    package ParseLink;
    our @ISA = qw(HTML::Parser);

    # called by parse
    sub start
      {
        my ($this, $tag, $attr) = @_;

        if ($tag eq "a")
        {
          # You might want to check for name="#anchor" links
          # here ...
          $this->{links}{$attr->{href}} = "(no text given)";
          $this->{curr_link} = $attr->{href};
          $this->{nesting_a}++;
        }
      }

    sub end
      {
        my ($this, $tag, $attr) = @_;

        if ($tag eq "a")
        {
          $this->{nesting_a}--;
          $this->{links}{$this->{curr_link}} = $this->{curr_text} 
            if $this->{curr_text};
        }
      }

    sub text {
      my ($this, $text) = @_;
      $this->{curr_text} .= $text if $this->{nesting_a} > 0;
    };

    sub get_links
      {
        my $this = shift;
        return keys %{$this->{links}};
      }
  }
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perl -MHTTP::Daemon -MHTTP::Response -MLWP::Simple -e ' ;    # The  
$d = new HTTP::Daemon and fork and getprint $d->url and exit;#spider
($c = $d->accept())->get_request(); $c->send_response( new   #in the
HTTP::Response(200,$_,$_,qq(Just another Perl hacker\n))); ' #  web
[download]