  The neutrino force results from the exchange of a pair of neutrinos. A
neutrino background can significantly influence this force. In this work, we
present a comprehensive calculation of the neutrino force in various neutrino
backgrounds with spin dependence taken into account. In particular, we
calculate the spin-independent and spin-dependent parity-conserving neutrino
forces, in addition to the spin-dependent parity-violating neutrino forces with
and without the presence of a neutrino background for both isotropic and
anisotropic backgrounds. Compared with the vacuum case, the neutrino background
can effectively violate Lorentz invariance and lead to additional
parity-violating terms that are not suppressed by the velocity of external
particles. We estimate the magnitude of the effect of atomic parity-violation
experiments, and it turns out to be well below the current experimental
sensitivity.
