my $fmt = '%d %b %Y %T %Z';

Don't use %Z as it may produce ambiguous time zone names like CST ("Central Standard Time", "China Standard Time", "Cuba Standard Time", ...), and as you can tell Time::Piece's strptime can't handle parsing its own output. Personally, I usually use DateTime::Format::Strptime for parsing, but AFAICT Time::Piece works with %z as well.

use warnings;
use strict;
use Time::Piece;

my $fmt = '%d %b %Y %T %z';  # %z instead of %Z appears to work
my $str = localtime->strftime($fmt);
print $str, "\n";
my $strtime = Time::Piece->strptime($str, $fmt);
print $strtime->strftime($fmt), "\n";
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use warnings;
use strict;
use DateTime;
use DateTime::Format::Strptime;

my $fmt = '%d %b %Y %T %z';

my $strp = DateTime::Format::Strptime->new( on_error=>'croak',
    pattern=>$fmt );

my $dt = DateTime->now;
$dt->set_time_zone('America/Chicago');  # try this with "%Z"...
print $dt->rfc3339, "\n";
my $str = $dt->strftime($fmt);
print "$str\n";
my $dt2 = $strp->parse_datetime($str);
print $dt2->rfc3339, "\n";
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