Why don't I get an chance to respond to the Extinction Event choice?
You cannot react to the even/odd choice in Extinction Event because that choice is made during the spell's resolution.
In general, you don't make all kinds of choices while casting a spell. As specified in rules 601.2b-d, you choose modes, splices, other cost modifiers, targets, and how to divide or distribute an effect among targets. Any other choice, like even/odd in Extinction Event, doesn't happen until the spell resolves.
While the choice on Extinction Event may look similar to a modal choice, a modal choice specifically requires you to choose from a list of actions, so an instruction to &quot;choose&quot; other things is different.

This fact is also mentioned in a Gatherer ruling on Extinction Event:

You choose odd or even while Extinction Event is resolving. Players can’t take any actions between the time you make that choice and the time some creatures go extinct.