Hello Charles,
That's not currently possible, as new items always default to their parent's dates and context. However, it's been requested by a few of us. I took it a step further and have proposed that it should be possible to select whether start and due dates and contexts get set to blank, the current date, or inherit from their parent or nearest sibling. I'll add your vote to the list, in the beta testers' forum.
Stéphane