The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order \(\beta\leq 2\). It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data v, \(v\in L_ 2\), optimal \(O(h^ r)\) convergence for positive time is possible in general only if \(r\leq 4-\beta\).