As dmckee wrote, the term "symmetry" has a fully uniform meaning. It is not used ambiguously in any way and for the same reason, it is not overused. Symmetries are really important in physics and that's why they're used so often. (We also use "symmetries" with various well-defined adjectives such as "global", "local/gauge", "approximate", "broken", "accidental", "discrete", "continuous", "internal", and so on.) I have to emphasize these things because this incorrect assumption of the OP is probably behind additional incorrect assumptions that lead to the questions which look so confusing to us.
It is hard to figure out what the question is (or questions are) actually about. It wants to suggest that P has to be automatically a symmetry because of some principles of special or general relativity. But this is not the case.
Only CPT has to hold because of similar symmetries: it may be interpreted as the rotation of the Euclideanized spacetime by $\pi$, so it follows from Lorentz symmetry. Assuming CPT, then symmetry under CT, TP, T is the same thing as symmetry under P, C, CP, respectively.
So there are three cases to discuss. The symmetry under P as well as under C may be broken with left-handed and right-handed spinors treated separately, as understood from the 1950s or so. Neutrinos are always left-handed, so there is no P image of them which would be light right-handed neutrinos. Antineutrinos are similarly right-handed and there is no P image, left-handed antineutrinos. So one could think that the combined symmetry CP, mapping genuine left-handed neutrinos to right-handed antineutrinos, could always be a symmetry.
At the level of field content, it is true in QFT that the spectrum is CP-symmetric. However, the full Lagrangian doesn't have to be. The CKM matrix violates CP in a way we know experimentally. Possibly, the QCD theta-angle could violate the CP as well except that it seems to vanish. SUSY and other models of new physics contain lots of new sources of CP violation.
It is not true that P has to be a symmetry because of the Lorentz symmetry. The Lorentz symmetry is SO(3,1) or Spin(3,1) - or the component of identity - which isn't the same thing as O(3,1), and therefore doesn't include the parity transformation. It's very natural to construct theories that are invariant under SO(3,1) but not O(3,1) and our world is an example. In the same way, one shouldn't automatically say that the laws of physics are invariant under the time reversal. When one does figure out what nontrivial "discrete" transformation automatically follows from the part of Lorentz symmetry that has to hold if the infinitesimal ones do, one only gets CPT. All other discrete symmetries of this kind may be broken and are broken in the real world, indeed.
Even in GR where one allows "all" coordinate transformations, one has to study which transformations preserve the local physics, and only SO(3,1), and not O(3,1), preserve the local physics. In particular, one may construct GR-like theories with spinors. They need "tetrads" which allow one to construct chiral spinors even in GR, and couple them to the metric etc. The outcome is, of course, that the physics is left-right-asymmetric, much like in SR. Only the diffeomorphisms "connected to the identity", and they don't include the parity, automatically follow from the invariance under infinitesimal diffeomorphisms.
Also, CPT (or T, in theories where it holds), only applies to the microscopic laws of Nature. It doesn't mean that the actual macroscopic processes or effective theories are time-reversal-invariant. They're surely not, the phenomena in our world are self-evidently irreversible: the second law of thermodynamics is the most well-known example of the inherent arrow of time that treats the past and future asymmetrically and can't ever be "revoked" in any way, by any genuine symmetry.
Just like the OP underestimates the importance of symmetries, he seems to underestimate the importance of the Lagrangians, too. He asks whether there is "anything else than just a property of the Lagrangians". But Lagrangians, whenever they exist, actually contain all the dynamical information about the laws of Nature. They're the whole thing. So saying that something is "only" about a property of the Lagrangian is missing the point of physics. Everything in science is just about a property of the Lagrangian.