In this paper we discuss the electric field component parallel to the magnetic field arising from plasma flows which violate the frozen-in field condition of ideal magnetohydrodynamics. The quantity of interest is the potential U = -integral E parallel-to ds where the integral is extended along field lines. It is shown that U can be directly related to magnetic field properties, expressed by Euler potentials, even when time-dependence is included. Our results are applicable to Earth's magnetosphere, to solar flares, to aligned-rotator models of compact objects, and to galactic rotation. On the basis of order-of-magnitude estimates, our results support the view that parallel electric fields associated with nonideal plasma flows might play an important role in cosmic particle acceleration.