Without addressing parallelization nor performing any benchmarks, here's naive implementation. Hopefully, fuzzy matching implemented and optimized in C is fast. Maybe, re-writing this with String::Approx instead of re::engine::TRE (subroutine call instead of Perl regexp engine overhead) would be faster. Re-visiting fuzzy string matching was fun :). Solution below was more readable/clear before I tried to get to (perceived, no tests) optimizations like dropping blocks, reversing loop, + remembering that to use a reference to substr result is efficient (no idea if it holds in this case), etc., but here's FWIW.

use strict;
use warnings;
use feature 'say';

use constant ERR => 1;
use re::engine::TRE 
    max_cost => ERR,
    cost_ins => -1,     # no insertions
    cost_del => -1,     # no deletions
;

my $suffix_source = 'abbaba';
my $prefix_source = 'babbaaaa';

my $max_len = 0;
$prefix_source =~ /^${ \substr $suffix_source, -$_ }/
    and $max_len = $_
    and last 
for reverse ERR + 1 .. length $suffix_source;

say substr( $suffix_source, -$max_len ),
    ' ',
    substr( $prefix_source, 0, $max_len )
if $max_len;

__END__

baba babb
[download]

Update: crude benchmarks (~200 chars strings, ~10 errors allowed) reveal that, for this task, both modules I mentioned are hugely (some 10s of times) slower than "classic" Perl implementation by tybalt89 :)