A parented prop is placed in relation to the body part its parented too. I dont remember offhand if its origin or center. *If the body part is morphed the position of the parented prop wont move. You can do a master/slave dial  to react to the morph channel and move the prop.

For things like watches there is the "adopt bends of parent" or somesuch which might be useful.

This keeps being discussed but there seems to be no real solution yet.   There were some scripts written to keep the prop at a specific vertex but they were a bit tedious to use.

It's inherit bends of parent, and it works quite well with things like armor and such.  It does not, however, work well with some weight mapped figures.  I do know that inherit bends does not work with Dusk, but I don't know how well it works with SM figures like Roxie and Miki4 as I have never tested it.