If you want to parse a file, use parse_file(). parse() is for parsing a html string. Use table_states(), not tables(), which is deprecated.

use Data::Dumper;
use strict;
use warnings;

use HTML::TableExtract;

my $te = HTML::TableExtract->new(headers => ['Name', 'Phone Number']);
 
$te->parse("<table><tr><td>Name</td><td>Phone Number</td></tr>"
           . "<tr><td>Tom</td><td>1234</td></tr>"
           . "<tr><td>Mary</td><td>4321</td></tr></table>");
foreach my $ts ($te->table_states) {
   print "Table (", join(',', $ts->coords), "):\n";
   foreach my $row ($ts->rows) {
      print join(',', @$row), "\n";
   }
 }
[download]

This gives:

Table (0,0):
Tom,1234
Mary,4321
[download]