I am confused with the parallel axis theorem, especially with the term $M$ in $Md^2$.
I don't understand why the moment of inertia of each particle increases by a term of $(dm)(d^2)$.
Is there a physical reason for why we should use the total mass of the rotating rigid body in the extra term $Md^2$?
Take a look at the picture above. In the first picture, the axis passes through the center of mass of the disc and the dashed line is the axis about which it will rotate.
When the axis changes as shown in the second figure, the molecules in the region $A$ are farther from the axis and hence the $mr^2$ term for them increased.
For the molecules in region $B$, the distance from the axis decreased and hence the $m'r^2$ term also decreased. And in region $C$, there apparently isn't any change at all.
So it is clear that the net change is not equal to $Md^2$.
Please correct me if I am wrong somewhere. And also here $r$ is any arbitrary distance.