How can I calculate a local rotation angle of subobjects along with the local y-axis with respect to the parent's object?

Not sure if I completly understand what you mean but to put it in my words:

You want to calculate the angle between the y-axis of transform A and the y-Axis of transform B, where B is the c$$anonymous$$ld of A?

If that is the case try t$$anonymous$$s:

   float angle = Vector3.Angle(transform.up, transform.parent.up);

(T$$anonymous$$s assumes that we calculate t$$anonymous$$s on the c$$anonymous$$ld transform) To put it more generalized:

   Transform c$$anonymous$$ldTrans = transform.Find("someC$$anonymous$$ld");
   float angle = Vector3.Angle(transform.up, c$$anonymous$$ldTrans.up);

Let me know if that helped..