It's not possible to parent the spawner to anything, there's simply no Parent parameter in there.
Creating one won't solve the problem, too. There's no code for it in the engine and Orangebox also seems to have a problem with func_door parented to a moving entity like Tracktrain.
It's also not possible to disable a ge_w/e-spawn entity in a map, otherwise i would suggest you to have the tracktrain activating the entity when it reached a certain pathnode and stoped there.
Ofc. it would have to be deactivated again if the elevator starts moving, otherwise the entity will simply relocate to it's spawn location and might block the elevator.
If you ask me, simply forget about this idea and return to the more vital things you have to take care of in your map.
I might have another idea soon, but first i need to check if it's working.