  Intrinsic quantum correlations supported by the $SU(2)\otimes SU(2)$
structure of the Dirac equation used to describe particle/antiparticle states,
optical ion traps and bilayer graphene are investigated and connected to the
description of local properties of Dirac bi-spinors. For quantum states driven
by Dirac-like Hamiltonians, quantum entanglement and geometric discord between
spin and parity degrees of freedom - sometimes mapped into equivalent low
energy internal degrees of freedom - are obtained. Such \textit{spin-parity}
quantum correlations and the corresponding nonlocal intrinsic structures of
bi-spinor fermionic states can be classified in order to relate quantum
observables to the (non)local behavior of these correlations. It is shown that
free particle mixed states do not violate the Clauser-Horne-Shymony-Holt
inequality: the correlations in such mixed bi-spinors, although quantum, can be
reproduced by a suitable local hidden variable model. Additionally, the effects
due to a non-minimal coupling to a homogeneous magnetic field, and to the
inclusion of thermal effects are evaluated, and quantum correlations of
associated quantum mixtures and of the thermal states are all quantified.The
above-mentioned correlation quantifiers are then used to measure the influence
of CP transformations on \textit{spin-parity} quantum correlations, and our
results show that quantum entanglement is invariant under CP transformations,
although the geometric discord is highly sensitive to the CP symmetry.
