• Do not use parse (), use getline ()
• Use a lexical file handle
• Never use an else branch if the if branch ends the enclosing scope (exit, return, die, last, …)
• Use auto_diag
• Use positive logic for validation

# Use auto_diag to see all parsing errors immediately
my $csv = Text::CSV_XS->new ({ binary => 1, auto_diag => 1, allow_whit
+espace => 1 });
open my $fh, "<", $file_to_parse or die "Could not open '$file' $!";
while (my $row = $csv->getline ($fh)) {

    # Check first and third field to be numeric only (and not empty)
    $row->[0] =~ m/^[0-9]+$/ && $row->[2] =~ m/^[0-9]+$/ or
        exit;        # wrong format

    # THERE CANNOT BE AN ELSE AFTER AN EXIT !
    }
[download]

update: your sample data contains spaces. You will have to deal with those, as the current tests will fail for leading and training spaces. Adding allow_whitespace option to the constructor will help you here.

POSIX [:alnum:] means alphanumeric You probably want [:digit:].

Maybe this is what you are taring to do.

#!/usr/bin/perl
# 
use strict;
use warnings;
use Text::CSV_XS;

my $csv = Text::CSV_XS->new();
#open (CSV, "<", $file_to_parse) or die "Could not open '$file' $!<br>
+";
while (my $line = <DATA>) {
    chomp $line;
    if ($csv->parse($line)) {
        my @columns = $csv->fields();
        $columns[0]  =~ s/ //g;
        $columns[2]  =~ s/ //g;
        if( ($columns[0]=~/^[0-9]+$/) && ($columns[2]=~/^[0-9]+$/) ) {
            print "Format ok\n"; foreach (@columns){print $_,"\t";};
            print "\n";
        }else{
            print "Wrong Format\n";
            print "First $columns[0] and tird $columns[2] have to be n
+umeric\n";
        }
        }else{
            my $err = $csv->error_input;
            print "<br><b>$err Failed to parse line, please call for h
+elp!</b><br>" if $err;
        }
}
__DATA__
12345,John Dow,           123456
54321,ANDREW MALL,        8888882
987865,BOGGS BUNNY,       2234566
88777,CLARK KENT R,         888666
126677,CLARK KENT R,     xx345778
112222,DAVOLIO DARK F,          3321777
[download]

Results

perl "E:\perl_TK\perlCSV.pl"
Process started >>>
Format ok
12345    John Dow    123456    
Format ok
54321    ANDREW MALL    8888882    
Format ok
987865    BOGGS BUNNY    2234566    
Format ok
88777    CLARK KENT R    888666    
Wrong Format
First 126677 and tird xx345778 have to be numeric
Format ok
112222    DAVOLIO DARK F    3321777    
<<< Process finished.
[download]

Your code currently will only throw an error if both columns are non-alphanumeric. If either one contains alphanumeric data, then it won't exit. You might want to change that "&&" to "||".

Give us a sample of your data...

Sample Data:

12345,John Dow,           123456
54321,ANDREW MALL,        8888882
987865,BOGGS BUNNY,       2234566
88777,CLARK KENT R,         888666
112222,DAVOLIO DARK F,          3321777
[download]

Still a problem with:

if( ($columns[0]!~/^[[:digit:]]+$/) || ($columns[2]!~/^[[:digit:]]+$/)
+ ) {
#stop the program here
}
[download]