The explanation is simple--- all particles we can see are chiral, they have only one handedness, because if they had both handedness, they could get a mass, and generically, that mass would be of order of magnitude the Planck mass. We live at energy scales which are teeny compared to the Planck mass, so we can only see massless stuff, so we only see chiral fermions (and gauge bosons).
The right question then is the other way around, if everything is chiral, why then do the electromagnetic and strong interactions not violate parity?
This is because the Higgs mechanism partners up the chiralities into massive Dirac particles at lower energies, and only the W,Z bosons know that they were chiral to begin with. At low energies, you get parity and charge-conjugation as accidental symmetries, because these are symmetries of the low-energy Dirac Lagrangian coupled to the remaining photon and gluons.
As for the neutrinos, a chiral neutrino can have a Majorana mass while only having one chirality, and this is certainly what is going on in nature, since this scheme predicts the mass correctly, and this mass is absurdly small.
This post imported from StackExchange Physics at 2014-07-18 05:01 (UCT), posted by SE-user Ron Maimon