In graphene we call the low energy excitations around the Dirac point Dirac fermions, which are massless. Is this just by convention or is there any further differences between massless Dirac fermions and Weyl fermions
Dirac fermions is only the direct sum of left- and right-handed Weyl representations (which leads to time inversion, charge inversion and spatial inversion invariance of the theory). Two Weyl representations are mixed by the mass term in the Dirac equation. If we set mass to zero, we will get two uncoupled equations, each of which describes Weyl fermion. But the Dirac theory remains the theory of direct sum of Weyl spinors even in massless case.
Yes, the parity symmetry can be implemented only for the four-component Dirac fields and not for two-component Weyl fields. This fact, mathematically speaking, does not depend on the value of the mass. Physically speaking, however, I am not sure that a massless four component spinor makes much sense in standard theories (Sorry, I do not know anything about graphene!).