  We prove the continuity of bounded solutions for a wide class of parabolic
equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla
u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized
non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant
c(K)\,g(y,\tau,{\rm v}/r), \quad (x,t), (y,\tau)\in Q_{r,r}(x_{0},t_{0}), \quad
0<{\rm v}\leqslant K\lambda(r), $$ $$ \quad
\lim\limits_{r\rightarrow0}\lambda(r)=0, \quad \lim\limits_{r\rightarrow0}
\frac{\lambda(r)}{r}=+\infty, \quad \int_{0} \lambda(r)\,\frac{dr}{r}=+\infty.
$$ In particular, our results cover new cases of double-phase parabolic
equations.
