We have measured parity-violating asymmetries in elastic electron-proton and quasielastic electron-deuteron scattering at Q2=0.22 and 0.63 GeV2. They are sensitive to strange quark contributions to currents in the nucleon and the nucleon axial-vector current. The results indicate strange quark contributions of approximately < 10% of the charge and magnetic nucleon form factors at these four-momentum transfers. We also present the first measurement of anapole moment effects in the axial-vector current at these four-momentum transfers.