Magnetically aligned velocity anisotropy over varying physical conditions and environments within the Taurus molecular cloud is evaluated from analysis of wide field spectroscopic imaging of 12CO and 13CO J= 1–0 emission. Such anisotropy is a result of magnetohydrodynamic turbulence in the strong magnetic field regime and provides an indirect measure of the role of magnetic fields upon the gas. Velocity anisotropy aligned with the local, projected mean magnetic field direction is limited to fields with low surface brightness 12CO emission corresponding to regions of low visual extinction and, presumably, low gas volume density. The more optically thin 13CO J= 1–0 emission shows little evidence for velocity anisotropy. We compare our results with computational simulations with varying degrees of magnetic field strength and Alfvénic Mach numbers. In the diffuse, molecular envelope of the cloud, a strong magnetic field and sub-Alfvénic turbulent motions are inferred. Super-Alfvénic motions are present within the high column density filaments of the Taurus cloud. From this trans-Alfvénic flow, we constrain the scaling exponent, κ, of the magnetic field density relation (B∼nκ) to be near zero as expected for ambipolar diffusion or material loading of magnetic flux tubes.