Can you email them directly and ask? That might be the best way to clear it up
I did email them, actually. But I think it went straight to Japan, where they had a bit of trouble translating it.

I emailed them, with a snippet of a quote from their own disclaimer pdf, and said that the disclaimer made it sound like we needed a release for any people pictures, even if there were pictures that we not normally need to have a release for. I asked if they could please tell me if that was accurate
I got a reply back that said, please reference our disclaimer document and then supplied the same quote that I had included in my original email. VERY helpful.
True it wouldn't be possible to get releases signed during a parade! - not unless you can ride along in a clown car. Seriously I suppose you'd have to be at the end of the parade and catch the subjects then or maybe contact the group that marched or sponsored the float etc.
In the disclaimer from the contest it says that entrants are required to obtain permission, and that entrants agree to accept responsibility for complaints etc. That seems to be saying that you need to get a release and if someone sees their photo used without their permission and complains it would be your responsibility to deal with the complaint.
ASMP of course is only one of many professional photographers organizations, but on their site the section about releases suggests getting a release anytime the subjects are recognizable in the photo, and to do so unless you're sure you'd never want to use the image for any purpose other than editorial. I think that retail use would include making money by selling your photos and would require a release.
Property and Model Releases | American Society of Media Photographers
Sharon
In the end, THIS was the kicker for me. The contest disclaimer document sounded as if it was saying that:
a) you're responsible for getting releases on any people photos, and
b) should anyone complain or sue, it's YOUR problem, not ours.
Whether they could legitimately get away with it being solely the photographer's problem if they were sued for using a photo, I don't know.
But they can certainly require releases as a condition for their contest, even when one might not ordinarily be necessary.
So, I did the only thing I could do to be absolutely certain there wasn't going to be a problem--I didn't enter the people photos. And I'm pretty sure the bugs and birds I entered won't sue me.

None of the people pictures were really anything outstanding anyway; I was just trying to toss some in there because it looks like past winners are heavily skewed toward people photos. But, it's really better to go with my strengths anyway.
I appreciate all the helpful comments though, and I think I'll be checking out some of those links, as I do plan to work on learning people photography a little better this year.