You can place your groups of data in an array. Your problem is in having several variables all the same name.

my @param_groups = (
    { x => 1, y => 2, z => 3 },
    { x => 3, y => 4, z => 5 },
    { x => 1, y => 1, z => 1 },
);
my ($count, %result) = 1;
for (@param_groups) {
    $result{'ParamGroup'.$count++}
        = add_them(@{$_}{qw/x y z/});
}
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