It is well known now that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly null-controllable in L(2)(Omega), when governed in a bounded domain by the locally distributed controls. In this paper we intend to show that, in fact, in one space dimension (Omega = (0,1)) the very same results can be achieved by employing at most two mobile point controls with support on the curves properly selected within an arbitrary subdomain of Q(T) = (0,1) x ( 0,T). We will show that such curves can be described by a certain differential inequality and the explicit examples are provided. We also discuss some extensions of our main results to the superlinear terms and to the case of several dimensions.