This will do it a bit more nicely (fewer branches) without modifying the variables (and using only if/elsif as requested):

use 5.010;

my ($x, $y, $z) = (-11,-13,4);

if ($x % 2 and ($x > $y or not $y % 2) and ($x > $z or not $z % 2)) {
    say "$x is the largest odd";
}
elsif ($y % 2 and ($y > $z or not $z % 2)) {
    say "$y is the largest odd";
}
elsif ($z % 2) {
    say "$z is the largest odd";
}
else {
    say "All numbers are even!";
}
[download]

Update: If I were doing this for real, I'd do:

my @nums = (-11, -13, 4);
my @candidates = sort { $a <=> $b } grep $_ % 2, @nums;
if (@candidates) {
    say $candidates[-1];
} else {
    say "They're all even!";
}
[download]