I wouldn't parent the prop to another prop. I would parent all props directly to the dummy character.
This is possible cause they have everytime the same relative positions to each other.
And so they could be deleted as single choice independently.

If you have all loaded once in right default position, you can save the complete thingie as one character file including props. This will load all with one click. I have done so with my V4 swordbelt and it's props.

And yes, conforming or moving that character will move also the parented props.
The props will not follow if you use a morph or scaling to change the character in it's size.

You are right TrekkieGrrrl opted for a prop. But I have understood that she wanted to avoid additional trouble in making the mask a character. Therefor I mentioned a dummy character.
Otherwise I would have opted to make the mask a cr2 instead of a dummy, though that's also not really complicated. It's only for one figure part and no big deal's with bending/welding some bodyparts.

Are you sure that a empty geometry will work fine as a character, even if only dummy.
I have done something with alternate geometry and have found that poser is sometimes very picky with a empty part only being a place holder in start position. If you than turn to a filled alternation and goes back the mesh will not disappear.