  We prove a local higher integrability result for the gradient of a weak
solution to parabolic double-phase systems of $p$-Laplace type when
$\tfrac{2n}{n+2}< p\le2$. The result is based on a reverse H\"older inequality
in intrinsic cylinders combining $p$-intrinsic and $(p,q)$-intrinsic
geometries. A singular scaling deficits affects the range of $q$.
