  In the presence of P-violating interactions, the exchange of vector bosons
between electrons and nucleons induces parity-nonconserving (PNC) effects in
atoms and molecules, while the exchange of vector bosons between nucleons
induces anapole moments of nuclei. We perform calculations of such
vector-mediated PNC effects in Cs, Ba$^+$, Yb, Tl, Fr and Ra$^+$ using the same
relativistic many-body approaches as in earlier calculations of standard-model
PNC effects, but with the long-range operator of the weak interaction. We
calculate nuclear anapole moments due to vector boson exchange using a simple
nuclear model. From measured and predicted (within the standard model) values
for the PNC amplitudes in Cs, Yb and Tl, as well as the nuclear anapole moment
of $^{133}$Cs, we constrain the P-violating vector-pseudovector
nucleon-electron and nucleon-proton interactions mediated by a generic vector
boson of arbitrary mass. Our limits improve on existing bounds from other
experiments by many orders of magnitude over a very large range of vector-boson
masses.
