Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations u′=A(u)u+f(u) is established. The focus lies on the case of strict inclusions dom(f)⊊dom(A) of the domains of the nonlinearities u↦f(u) and u↦A(u). Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.