We consider a system of singularly perturbed differential equations with singular parameter ε<<1, discretized with an IMEX Runge-Kutta method. The splitting needed for the IMEX method stems from a linearization of the fluxes around the limit solution. We analyze the asymptotic convergence order as ε→0. We show that in this setting, the minimal stage order of the implicit part of the scheme is of great importance, thereby explaining earlier numerical results showing a close correlation of errors of the splitting scheme and the fully implicit one.