In this paper we will focus on numerical methods for differential equations with both stiff and nonstiff parts. This kind of problems can be treated efficiently by implicitexplicit (IMEX) methods and here we investigate a class of s-stage IMEX peer methods of order p = s for variable and p = s+ 1 for constant step sizes. They are combinations of s-stage superconvergent implicit and explicit peer methods. We construct methods of order p = s + 1 for s = 3, 4, 5 where we compute the free parameters numerically to give good stability with respect to a general linear test problem frequently used in the literature. Numerical comparisons with two-step IMEX Runge-Kutta methods confirm the high potential of the new constructed superconvergent IMEX peer methods.