The aim of this paper is to give a simple proof of the existence of a smooth solution to the semilinear parabolic equation with fourth order elliptic operator: \[ u_ t= -\varepsilon^ 2 \Delta^ 2 u+ f(t,x,u,u_ x,u_{xx}),\tag{1} \] \(x\in \Omega\subset \mathbb{R}^ n\), \(\Omega\) is a bounded domain, \(t\in [0,T_{\max})\), \(T_{\max}\leq +\infty\). We consider (1) together with initial-boundary conditions \[ u(0,x)= u_ 0(x),\quad x\in \Omega,\quad {\partial u\over\partial n}= {\partial(\Delta u)\over \partial n}=0\quad\text{when } x\in \partial\Omega. \] The general scheme of our proof of local existence is similar to the classical proof of the Picard theorem for solutions of ordinary differential equations.