The regex doesn't remove the dot because the regex doesn't match. Let's make two small changes:

#!/usr/local/bin/perl -w

$a = "$$NETOBJ.parameter";
print ">>$a<<\n";
$a =~ s/\$\$NETOBJ\.//;

print $a;
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You might also write it this way:

$a = "$$NETOBJ.parameter";

if ($a =~ s/\$\$NETOBJ\.//) {
    print $a;
} else {
    print "Substitution failed.\n";
}
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One other approach that may work for you is split:

$a = '$$NETOBJ.parameter';
(undef, $a) = split(/\./, $a, 2);
print $a;
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Of course, notice the single quotes in the declaration that time. That'll help even if you do use the regexp.