Parents usually being a little suspicious of their offspring, I would expect the alarm bells to go off when they realise they are in club from NCL to LHR. Plus there is always the risk of one of the members of staff trying to be helpful, e.g. by pointing out that "As you are traveling first, you'll have access to the CCR in LHR".

Have fun in NYC - it's great to be able to spoil the parents every once in a while...