Consider a body orbiting at some distance around a fixed point and spinning about its center of mass (such that both rotations are in the same plane). 

The orbital angular momentum will be that of a point particle positioned at the center of mass: $$L_{orbit} = mr_{CM}^2\omega_{orbit}$$

The spin angular momentum will be:
$$L_{spin} = I_{CM}\omega_{spin}$$

So the total angular momentum is:
$$L = mr_{CM}^2\omega_{orbit} + I_{CM}\omega_{spin}$$

Now try to picture that a body rotating at angular frequency $\omega$ about a point a distance $d$ from its center of mass is nothing but the previously considered system, with that point as the center of orbit and where the orbit is in "tidal lock", i.e. $r_{CM} = d$ and $\omega_{orbit} = \omega_{spin} = \omega$. The angular momentum is thus

$$L = md^2\omega + I_{CM}\omega = (md^2+I_{CM})\omega$$

And hence we arrive at the parallel axis theorem:

$$I = md^2 + I_{CM}$$