  We investigate numerically parametrically driven coupled nonlinear
Schrodinger equations modelling the dynamics of coupled wavefields in a
periodically oscillating double-well potential. The equations describe among
other things two coupled periodically-curved optical waveguides with Kerr
nonlinearity or horizontally shaken Bose-Einstein condensates in a double-well
magnetic trap. In particular, we study the persistence of equilibrium states of
the undriven system due to the presence of the parametric drive. Using
numerical continuations of periodic orbits and calculating the corresponding
Floquet multipliers, we find that the drive can (de)stabilize a continuation of
an equilibrium state indicated by the change of the (in)stability of the orbit.
Hence, we show that parametric drives can provide a powerful control to
nonlinear (optical or matter wave) field tunneling. Analytical approximations
based on an averaging method are presented. Using perturbation theory the
influence of the drive on the symmetry breaking bifurcation point is discussed.
