Does it have to be a regex? tr/// provides a simpler solution.

#!/usr/bin/perl -w
use strict;
while (my $str = <DATA>) {
    my $odd = $str =~ tr/1// & 1;
    print $str,"has an ",$odd?"odd":"even"," number of 1's",$/;
}
__DATA__
110101010110111010101
100000000000000000001
101010101010101010101
110000000000000000011
110100000000000000000
11
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Output:

110101010110111010101
has an odd number of 1's
100000000000000000001
has an even number of 1's
101010101010101010101
has an odd number of 1's
110000000000000000011
has an even number of 1's
110100000000000000000
has an odd number of 1's
11
has an even number of 1's
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