Of course you can use HTML::Parser and code the table parsing by hand, but I suggest using HTML::TableContentParser, which is a subclass of HTML::Parser:

use strict;
use HTML::TableContentParser;

my $html = qq{
    <table>
        <tr><td>1</td><td>2</td><td>3</td></tr>
        <tr><td>4</td><td>5</td><td>6</td></tr>
        <tr><td>7</td><td>8</td><td>9</td></tr>
    </table>

    <table>
        <tr><td>11</td><td>12</td><td>13</td></tr>
        <tr><td>14</td><td>15</td><td>16</td></tr>
        <tr><td>17</td><td>18</td><td>19</td></tr>
    </table>
};

my $p = HTML::TableContentParser->new();

my $tables = $p->parse($html);

for my $table (@$tables)
{
    print "new table!\n";

    for my $row (@{$table->{rows}})
    {

        print "new row: ";
        for my $column (@{$row->{cells}})
        {
            print "[$column->{data}] ";
        }
        print "\n";
    }
}
[download]

That prints:

new table!
new row: [1] [2] [3] 
new row: [4] [5] [6] 
new row: [7] [8] [9] 
new table!
new row: [11] [12] [13] 
new row: [14] [15] [16] 
new row: [17] [18] [19]
[download]

Easy, nice, reliable and clean. Enjoy! ;-)