This is a tough one but we have a beautiful theorem called the Parallel-axis Theorem. In short it states that if we know the moment of inertia \(\displaystyle (I_{z})\) of a body through its center, then the moment of inertia through an axis parallel to the body's axis is:
\(\displaystyle I = I_{z} + Mh^2\)
where \(\displaystyle h\) is the distance between the two axes.
An axis tangent to the edge of the cylinder has a distance \(\displaystyle h = R\) to the center of the cylinder. And from a previous post we know that the moment of inertia of a uniform cylinder through its center is \(\displaystyle I_z = \frac{1}{2}MR^2\).
Then, the answer to this problem is:
\(\displaystyle I = I_{z} + Mh^2 = \frac{1}{2}MR^2 + MR^2 = \textcolor{blue}{\frac{3}{2}MR^2}\)
