  The nonlinear (full-$f$) electromagnetic gyrokinetic Vlasov-Maxwell equations
are derived in the parallel-symplectic representation from an Eulerian
gyrokinetic variational principle. The gyrokinetic Vlasov-Maxwell equations are
shown to possess an exact energy conservation law, which is derived by Noether
method from the gyrokinetic variational principle. Here, the gyrocenter Poisson
bracket and the gyrocenter Jacobian contain contributions from the perturbed
magnetic field. In the full-$f$ formulation of the gyrokinetic Vlasov-Maxwell
theory presented here, the gyrocenter parallel-Amp\`{e}re equation contains a
second-order contribution to the gyrocenter current density that is derived
from the second-order gyrocenter ponderomotive Hamiltonian.
