  We consider linear inhomogeneous non-autonomous parabolic problems associated
to sesquilinear forms, with discontinuous dependence of time. We show that for
these problems, the property of maximal parabolic regularity can be
extrapolated to time integrability exponents $r\neq 2$. This allows us to prove
maximal parabolic $L^r$-regularity for discontinuous non-autonomous
second-order divergence form operators in very general geometric settings and
to prove existence results for related quasilinear equations.
