We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $$extract_itex$$A$$/extract_itex$$ with non-standard growth controlled by an $$extract_itex$$N$$/extract_itex$$-function depending on time and spatial variable. We do not assume continuity in time for the $$extract_itex$$N$$/extract_itex$$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $$extract_itex$$N$$/extract_itex$$-function, we also prove their uniqueness. This result is based on the approximation theory in Bochner-Orlicz-Museliak-Sobolev spaces. Due to the structure of the problem, we obtain and additional regularization in time variable directly from the equations, which allows us to consider rather general-in-time $$extract_itex$$N$$/extract_itex$$-function. This general result applies to equations studied in the literature like $$extract_itex$$p(t,x)$$/extract_itex$$-Laplacian and double-phase problems. Moreover, we show, how the developed approximation method can be used for  the elliptic case to avoid the so-called Lavrientev phenomenon.