#!/usr/bin/env perl

use strict;
use warnings;

my $re = qr{^(.*)(?<![\\])[\\]\n$};

my $line = '';

while (<DATA>) {
    if (/$re/) {
        $line .= $1;
        next;
    }

    $line .= $_;
    print $line;
    $line = '';
}

__DATA__
Line 1 Part A \
Line 1 Part B \
Line 1 Part C
Line 2 ALL
Line 3 Part X \
Line 3 Part Y
Line 4 END WITH BACKSLASH \\
Line 5 LAST Z