Your input isn't correct MIME message. There should be header that defines boundary and it's missed. Here's some small and dirty (sorry...) example:

use strict;
use warnings;
use 5.010;
use MIME::Lite;
use Email::MIME;

my $msg = MIME::Lite->new(
    From => 'src@example.com',
    To   => 'dst@example.com',
    Subject => 'message',
    Type => 'multipart/alternative',
);
$msg->attach(
    Type => 'text/plain',
    Data => 'this is text content',
);
$msg->attach(
    Type => 'text/html',
    Data => 'this is <b>html</b> content',
);

my $msg_str = $msg->as_string;
print $msg_str; # note the output here -- that's a complete message

my $parsed = Email::MIME->new($msg_str);

say "*" x 50;
if ($parsed->content_type =~ m{^multipart/alternative}) {
    say get_text_parts($parsed)->body;
}

sub get_text_parts {
    my @parts = shift->parts;
    my %ct;

    $ct{$_->content_type} = $_ for @parts;
    return $ct{'text/plain'} if exists $ct{'text/plain'};
    return $ct{'text/html'} if exists $ct{'text/html'};
    return $parts[0];
}
[download]

Upd: minor fix

Upd: Note also that $_->content_type may return something like text/plain; charset=utf-8, and this code will fail in this case.

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