Diffusion problems where the current state depends upon an earlier one give rise to parabolic equations with delay. The efficient numerical solution of classical parabolic equations can be accomplished via methods for stiff differential equations; one such class is predictor-corrector-type methods with extended real stability intervals and with reduced storage requirements. Analogous methods for equations with delay are proposed and analysed here. Our analysis will be based on the test equation

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... Y(t)=q1y(t)+q2 y(t-w), where, in view of the class of parabolic delay equations we want to consider, our main interest will be in the case \q 1 \ » \q2\. Implementational details of the methods developed are given and numerical results are presented.