We study if the parabolic forward-in-time maximal operator is bounded on parabolic (Formula presented.). It turns out that for non-negative functions the answer is positive, but the behavior of sign-changing functions is more delicate. The class parabolic (Formula presented.) and the forward-in-time maximal operator originate from the regularity theory of nonlinear parabolic partial differential equations. In addition to that context, we also study the question in dimension one.