The property of maximal \(L_p\)-regularity for parabolic evolution equations is investigated via the concept of \(\mathcal R\)-sectorial operators and operator-valued Fourier multipliers. As application, we consider the \(L_q\)-realization of an elliptic boundary value problem of order \(2m\) with operator-valued coefficients subject to general boundary conditions. We show that there is maximal \(L_p\)-\(L_q\)-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.