  We prove the local H\"older regularity of weak solutions to the mixed local
nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta
u+\text{P.V.}\int_{\mathbb{R}^{n}}
{\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where
$0<s<1$; for every exponent $\alpha_0\in(0,1)$. Here, $\Delta$ is the usual
Laplace operator. Next, we show that the gradients of weak solutions are also
$\alpha$-H\"older continuous for some $\alpha\in (0,1)$. Our approach is purely
analytic and it is based on perturbation techniques.
