There are quite a few assumptions being made so it leads to excess code, especially since I don't have much time to really focus and golf it down, but give this a whirl:

use warnings;
use strict;

my $line1 = "Positive for Depression (  mother ; sister  ),  Type 2 Di
+abetes (  father ; mother ; grandparents  ) and  Anxiety ( mother) ."
+;
my $line2 = "Cancer ( mother, grandmother )";

my %issues;

for my $line (($line1, $line2)){
    $line =~ s/(?:Positive for | and )//g;
    $line =~ s/,/;/g;

    # remove a semi-colon, if it immediately follows a closing
    # parens. This ; is the comma after "; sister )" that we
    # turned into a semi-colon

    $line =~ s/(?<=\));//g;

    if (my @entries = $line =~ /.*?\(.*?\)/g){
        for (@entries){
            /(.*)\((.*)\)/;
            my $issue = $1;
            my $people = $2;
            $issue  =~ s/^\s+//g; # squash leading whitespace
            $people =~ s/\s+//g;  # squash all whitespace
            @{ $issues{$issue} } = split /;/, $people;
        }
    }
}

for my $issue (keys %issues){
    for my $person (@{ $issues{$issue} }){
        print "$person || $issue\n";
    }
}

__END__
mother || Cancer 
grandmother || Cancer 
father || Type 2 Diabetes 
mother || Type 2 Diabetes 
grandparents || Type 2 Diabetes 
mother || Depression 
sister || Depression 
mother || Anxiety
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