Below, a version using shared variables ($r, $g, $b) provided by threads::shared. This completes in 9.730 seconds.

use strict;
use warnings;

use threads;
use threads::shared;

use MCE::Flow;

my ($m, $n, $count_max) = (500, 500, 2000);

my ($x_max, $x_min, $y_max, $y_min) = (
   1.25, -2.25, 1.75, -1.75
);

my $r = shared_clone [];
my $g = shared_clone [];
my $b = shared_clone [];

for my $i ( 0 .. $m - 1 ) {
   $r->[$i] = shared_clone [];
   $g->[$i] = shared_clone [];
   $b->[$i] = shared_clone [];
}

#--------------------------------------------------------------------#

# Perl version of Mandelbrot_OpenMP by John Burkardt.
# http://people.sc.fsu.edu/~jburkardt/c_src/mandelbrot_openmp/
#      mandelbrot_openmp.html
#
# Carry out the iteration for each pixel, determining count.

sub mandelbrot
{
   my ($m1, $m2) = @_;
   my ($x, $x1, $x2, $y, $y1, $y2);
   my ($c, %count, $i, $j, $k);

   for $i ( $m1 .. $m2 ) {
      for $j ( 0 .. $n - 1 ) {
         $x = ( (      $j - 1 ) * $x_max
              + ( $m - $j     ) * $x_min )
              / ( $m      - 1 );

         $y = ( (      $i - 1 ) * $y_max
              + ( $n - $i     ) * $y_min )
              / ( $n      - 1 );

         $count{$i}{$j} = 0;

         $x1 = $x;
         $y1 = $y;

         for $k ( 1 .. $count_max ) {
            $x2 = ( $x1 * $x1 ) - ( $y1 * $y1 ) + $x;
            $y2 = 2 * $x1 * $y1 + $y;

            if ( $x2 < -2.0 || 2.0 < $x2 || $y2 < -2.0 || 2.0 < $y2 )
            {
               $count{$i}{$j} = $k;
               last;
            }
            $x1 = $x2;
            $y1 = $y2;
         }

         if ( ( $count{$i}{$j} % 2 ) == 1 ) {
            lock $r; # not necessary, but runs faster with locking
            $r->[$i][$j] = 255;
            $g->[$i][$j] = 255;
            $b->[$i][$j] = 255;
         }
         else {
            lock $r; # ditto
            $c = int(
               255.0 * sqrt(sqrt(sqrt($count{$i}{$j} / $count_max)))
            );
            $r->[$i][$j] = $g->[$i][$j] = 3 * $c / 5;
            $b->[$i][$j] = $c;
         }
      }
   }

   return;
}

#--------------------------------------------------------------------#

# Return the smaller of two numbers.

sub min
{
   $_[0] < $_[1] ? $_[0] : $_[1];
}

# Write image to an ASCII PPM file.

sub write_to_ppm
{
   my ($ofile) = @_;
   my ($FH, $i, $j, $jlo, $jhi);

   open  $FH, '>', $ofile or die "cannot open file ($ofile): $!\n";

   print $FH "P3\n";
   print $FH "$n  $m\n";
   print $FH "255\n";

   for ( $i = 0; $i < $m; $i++ ) {
      for ( $jlo = 0; $jlo < $n; $jlo += 4 ) {
         $jhi = min( $jlo + 4, $n );
         for ( $j = $jlo; $j < $jhi; $j++ ) {
            printf $FH "  %d  %d  %d",
               $r->[$i][$j], $g->[$i][$j], $b->[$i][$j]
         }
         print $FH "\n";
      }
   }

   close $FH;
}

#--------------------------------------------------------------------#

# The bounds_only option applies to sequence of numbers
# which means to compute the begin and end boundaries only,
# not the numbers in between. Thus, workers receive 2
# numbers in @{ $chunk_ref }.

# Same as mce_flow_s { mce options }, sub { ... }, 0, $m - 1;

MCE::Flow::run_seq( {
   chunk_size => 5, max_workers => 'auto', bounds_only => 1
}, sub {
   my ($mce, $chunk_ref, $chunk_id) = @_;
   mandelbrot( @{ $chunk_ref} );
}, 0, $m - 1 );

## Write image to an ASCII PPM file.

write_to_ppm('mandelbrot.ppm');
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