  This article focuses on parabolic equations with rough diffusion coefficients
which are ill-posed in the classical sense of distributions due to the presence
of a singular forcing. Inspired by the philosophy of rough paths and regularity
structures, we introduce a notion of modelled distribution which is suitable in
this context. We prove two general tools for reconstruction and integration, as
well as a product lemma which is tailor made for the reconstruction of the
rough diffusion operator. This yields a partially automated deterministic
theory, which we apply to obtain an existence and uniqueness theory for
parabolic equations with rough diffusion coefficients and a singular forcing in
the negative parabolic H\"{o}lder space of order larger than $-\frac{3}{2}$.
