Since you are going on your second cruise, you are silver castaway club and can book activities for people in your stateroom at 90 days before sailing. Sounds like your parents are on their first cruise with
DCL so their booking window doesn't open until 75 days before sailing. Your reservations may be linked but that does not extend your Castaway Club benefits to the additional stateroom. So when you go in on June 1st to book your activities, you will see your parents names but they will be grayed out (not selectable) until their booking window opens on June 16th. So your options would be to book activities for your stateroom on June 1st and then on June 16th try to add your parents to the same activities. Or to wait until June 16th and book everyone at the same time.
I guess there is another option as well but I always think it's more effort than it is worth. That's to change names on the reservation and have someone who has sailed before in each cabin. That would make both cabins Silver level and able to book activities on June 1st. Once on board, sleeping arrangements can be changed to how you need them and you can either have a separate KTTW card to the other cabin or have the KTTW card keyed to the cabin you are actually going to sleep in.