In this paper we consider parabolic Q-quasiminimizers related to the p-Laplace equation in Omega(T) : = Omega x (0, T). In particular, we focus on the stability problem with respect to the parameters p and Q. It is known that, if Q -> 1, then parabolic quasiminimizers with fixed initial-boundary data on Omega(T) converge to the parabolic minimizer strongly in L-p(0, T; W-1,W-p(Omega)) under suitable further structural assumptions. Our concern is whether or not we can obtain even stronger convergence. We will show a fairly strong stability result.