Its got something to do with $2 being a scalar, and being able to contain only a single value -- try the following with  -Mre=debug

$ perl -le " $_ = q{abcdef}; print for m{(.)(.)}g;
a
b
c
d
e
f

$ perl -le " $_ = q{abcdef}; print for m{(.)(.)*}g;
a
f
[download]

So I think the (old) idiom is this

$cmd=q|&COMPAREEQUAL(First-param.one, Second.param,Third-param)|;

my $re = qr{
\G
(?:
        \&
    |
        (\w+) # $1
        \(
    |
        (?:
            ( # $2
              [^,\)\s]+ 
            )
            |
            (?:
                [,\s]+
            )
        )
    |
        \)
)
}mx;

while( $cmd =~  m{$re}g ){
    use Data::Dump; dd [ $1, $2 ];
}

__END__
[undef, undef]
["COMPAREEQUAL", undef]
[undef, "First-param.one"]
[undef, undef]
[undef, "Second.param"]
[undef, undef]
[undef, "Third-param"]
[undef, undef]
[download]

Double-undef is when the non $1/$2 parts match