We analyze the linear polarization of the relativistic jet in BL Lacertae object OJ 287 as revealed by multi-epoch Very Long Baseline Array images at 43 GHz and monitoring observations at optical bands. The electric-vector position angle of the optical polarization matches that at 43 GHz at locations that are often in the compact millimeter-wave "core" or, at other epochs, coincident with a bright, quasi-stationary emission feature ∼0.2 mas (∼0.9 pc projected on the sky) downstream from the core. This implies that electrons with high enough energies to emit optical synchrotron and γ-ray inverse Compton radiation are accelerated both in the core and at the downstream feature, the latter of which lies ≥10 pc from the central engine. The polarization vector in the stationary feature is nearly parallel to the jet axis, as expected for a conical standing shock capable of accelerating electrons to GeV energies.