Why are you using a rig and bones at all, a telescope is is a hard object, you can use parenting and constraints, then use the transforms to do any animation of it you need. There isn't a need to use bones at all for it.
It's just a vertex weight problem, I think, perhaps because 'Automatic Weights' doesn't cut it, here.
If the vertices illustrated above are assigned with weight 1 to the groups tri_R, main, and tri_L respectively, everything seems to work.
Maybe select vertices in Edit Mode in a wireframe view, to ensure you get them all, and assign to groups with weight ...
To commincate the problem better I will concentrate on this simplified representation.
I will refer to the blue object as the Yaw, which will rotate around the Z axis and to the red object as the Pitch which will rotate around the X axis.
When rotating the Yaw object, the Pitch object should follow as if it was connected. When rotating the Pitch object, the ...
Maybe you are talking about this method but I don't know if there's another one. Let say you want to mirror the left arm group to the right.
If the right group already exists (because you've parented With Automatic Weight), delete it.
Select the left group and copy it.
Now mirror that copy.
Give it the correct name, i.e. with the correct suffix. This way ...
So what you could do is:
First, to clean a bit, remove the current armature and the object Armature modifier.
Also, delete all the existing vertex groups.
Prepare the right side of the armature, each bone must be the child of another one, the main one is the root bone. Put their head on the paper fold so that it will fold correctly. Give them some logical ...
Per my comment above, I've been trying to think about how best to achieve the same thing: introducing radical modularity into my models. I tried a few ways of precisely linking subsets of vertices across multiple objects with Hooks, Constraints, Vertex Parenting, etc, and found them all to be insufficiently flexible, and generally just very clearly not ...