Script:

#!/usr/bin/env perl
use strict;
use warnings;
use diagnostics;
use feature 'say';

while (<DATA>) {
    chomp;
    if ($_ ne q{}) {
        s/Positive\sfor\s//;
        while (/
            (?:and\s)?
            ((?:\w+\s)+) # Remember disease name
            \(
                ([^\)]+) # Remember list of family members
            \)
        /xg) {
            my ($disease, @persons) = ($1, map {
                s/^\s+//;
                s/\s+$//;
                $_;
            } split /;|,/, $2);
            $disease =~ s/\s$//;
            foreach my $person (@persons) {
                say "$person || $disease";
            }
        }
    }
}
close DATA
    or die 'Could not close DATA: ', $!;

__DATA__

Positive for Depression (  mother ; sister  ),  Type 2 Diabetes (  fat
+her ; mother ; grandparents  ) and  Anxiety ( mother) .

Cancer ( mother, grandmother )
[download]

Output:

mother || Depression
sister || Depression
father || Type 2 Diabetes
mother || Type 2 Diabetes
grandparents || Type 2 Diabetes
mother || Anxiety
mother || Cancer
grandmother || Cancer
[download]