I would actually cancel that reservation and call Hhonors and ask them to make a gift awards reservation. They will make it in the names of your parents. I think that with all your info on the reservation, that even if you add their name(s) it could lead to problems and confusion on check-in. International hotels are much more strict than US hotels on seeing ID and etc. Some Asian hotels (especially Japan/Singapore) are very very formal and protocol oriented.

On a different but related topic. All rewards reservations that I have made have had my diamond benefits honored...

FDW