General Relativity is intrinsically parity conserving. A change of coordinates under parity conserves all the equations of general relativity. At a more fundamental level, Spacetime is described as a manifold and mathematical results have to be valid for any coordinate system, including left handed and right handed ones.
If General relativity can explain particle properties, then they would be expected to conserve parity if the boundary conditions conserve parity, as they do. Yet the Weak interaction maximally violate parity - or so we are told.
Deeper investigation reveals that the experimental fact of parity violation in weak interactions is a mixture of experimental results, conventions and assumptions. Equally valid assumptions lead to the contrary conclusion that Parity is conserved by the weak interactions! Parity is most sensibly represented by the CP operator. See my eprint for a discussion. In brief: it is equally valid to regard particles and intrinsically asymmetric but that weak interactions, like all other interactions, conserve parity
However there seems to be clear experimental evidence for CP violation (that I would call the real parity operation). This is not well understood. CP violation can be accounted for by a complex phase in CKM matrix, but this falls a long way short of an explanation - it just helps calculate the reaction rates. If my work is correct then there must be an explanation for the CP violation results, seen in the neutral Kaons and the B factories, in terms of some external factor rather than as an intrinsic property of elementary particles. A paper in preparation gives just such an explanation.