HTML::TokeParser::Simple is a good HTML parser to get started with and the docs are friendly. :-)

I'm not to sure about your spec but if it is get the text within bold tags that are within td tags then something like this will get you started.

#!/usr/bin/perl

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

my $html = do{local $/;<DATA>};
my $p = HTML::TokeParser::Simple->new(string => $html);

my ($in_td, $in_b);
while (my $t = $p->get_token) {
  $in_td++, next if $t->is_start_tag(q{td});
  $in_b++,  next if $in_td and $t->is_start_tag(q{b});
  next unless $in_td and $in_b;
  if ($t->is_text){
    print $t->as_is, qq{\n};
    $in_td = 0;
    $in_b = 0;
  }
}

__DATA__
<TR class="violet3">
  <TD ><B>hsa-miR-107</B></TD>
  <TD >17.1922</TD>
  <TD >-21.47</TD>
  <TD >2.119850e-02</TD>
  <TD >2.097540e-02</TD>
  <TD >6.191350e-04</TD>
  <TD >106</TD>
  <TD >127</TD>
  <TD ><pre><FONT COLOR="#FFFFFF">a</FONT><FONT COLOR="#FFFFFF">c</FON
+T><FONT COLOR="#FFFFFF">u</FONT><FONT></FONT>
</TR>
<TR class="violet2">
  <TD ><B>hsa-miR-103</B></TD>
  <TD >17.1922</TD>
<TR class="violet3">
  <TD ><B>hsa-miR-651</B></TD>
  </TR>
<TR class="violet2">
  <TD ><B>hsa-miR-320</B></TD>
[download]

hsa-miR-107
hsa-miR-103
hsa-miR-651
hsa-miR-320
[download]

Good luck!