I have an object with a number of geo objects that are parented to each other. When I bring that asset into Maya, the controls all work, but the parenting structure seems to be completely flattened out.

Is there some way to keep the parenting in the Maya instance of the asset?

Unfortunately, there's no way to preserve the parenting at the moment. I have a small explanation in this post:
https://www.sidefx.com/index.php?option=com_forum&Itemid=172&page=viewtopic&p=195083#195083 [sidefx.com]