Ken:
I find that parts get lost mostly either during construction, when I might drop one and it's never seen again, or, between the time I detach them from the sprue and the time I paint and assemble.
In the first case, I have no real solutions, except to tell myself not to be so ham-fisted and clumsy.
But in the second, I store parts until needed in plastic boxes that originally contained lunch meat. I don't remember what brands, but some thin-sliced lunch meat comes in clear plastic boxes with snap on lids.
Over the past year or two, I've saved them, and now have enough to devote one box to the parts for each construction step for most kits. It keeps them all together, and since the lids snap on, even if you knock one over, the parts stay inside.
Plus, they are FREE!!!! [after you pay for the meat]
Here's a photo of one, with a Hurricane sitting on it, to give you an idea of what to look for. The lid is not attached.
Since the boxes are clear, you can also see what you got pretty easily.
Sort of along Gary's line of thinking, I've sometimes stored the tiny parts in small sandwich bags with the zip lock strip.
Of course, if any cats are involved in mysterious disappearances of parts, I don't have suggestions to help that, either--at least, no suggestions that the ASPCA would approve of.