The first formula expresses the fact that for a system of particles, the total angular momentum of the system can be written as the angular momentum of the center of mass plus the total angular momentum due to the motions of the particles about the center of mass. See here for more details.
The second formula says that if you know the moment of inertia of a rigid body for rotations about an axis that passes through the center of mass then you can compute the moment of inertia of that same rigid body for rotations about any other axis that is parallel to it by adding a term that is like the moment inertia of a point particle of the same mass as the object.
They are conceptually distinct. The proofs of the parallel axis theorem that I have encountered do not make use of the first expression.