  In this paper, we obtain $C^{1,\alpha}$ estimates for weak solutions of
certain quasilinear parabolic equations satisfying nonstandard growth
conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2}
\nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla
u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori
have bounded gradient. We build on the recently developed scaling and covering
argument which allows us to consider the singular and degenerate cases in a
uniform manner and with minimal regularity requirements on the phase switching
factor $a(t)$ and the variable exponent $p(t)$. Moreover, we are able to take
any $p \leq q < \infty$ to obtain the desired regularity.
