= Solution
Choose parallel axes along $z$, one through the <centre of mass> and one displaced transversely by $\mathbf a$. Write the perpendicular position as $\mathbf r_\perp=\mathbf r'_{\perp}+\mathbf a$. Direct expansion of the <moment of inertia> gives
$$
I=\int|\mathbf r'_\perp+\mathbf a|^2\,dm=I_G+2\mathbf a\cdot\int\mathbf r'_\perp\,dm+ma^2.
$$
The integral in the cross term is zero by the definition of the <centre of mass>. Therefore the <parallel axis theorem> is
$$
\boxed{I=I_G+ma^2.}
$$
The axes must be parallel and the reference axis must pass through the <centre of mass>; the formula is not generally valid between arbitrary parallel axes using either axis as the reference.
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