For densities above n=1.6x10(11) cm(-2) in the strongly interacting system of electrons in two-dimensional silicon inversion layers, excellent agreement between experiment and the theory of Zala, Narozhny, and Aleiner is obtained for the response of the conductivity to a magnetic field applied parallel to the plane of the electrons. The parameters deduced from fits to the magnetoconductance do not provide quantitative agreement with the observed zero-field temperature dependence. We attribute this to the neglect in the theory of additional scattering terms, which affect the temperature dependence more strongly than the field dependence.