We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable:∂2u-P(a(u)∂12u+σ(u)f)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\partial_2u -P( a(u)\\partial_1^2u + \\sigma(u)f ) =0,$$\\end{document}where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of { C^{alpha-2}} for {alpha > frac{2}{3}} on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator {(1 + |partial_1|)^{-lambda_1 }} with {lambda_1 > frac13} is admissible. On the deterministic side we obtain a {C^alpha}-estimate for u, assuming that we control products of the form {vpartial_1^2v} and vf with v solving the constant-coefficient equation {partial_2 v-a_0partial_1^2v=f}. As a consequence, we obtain existence, uniqueness and stability with respect to {(f, vf, v partial_1^2v)} of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product {sigma(u)f} via a space-time version of Gubinelli’s notion of controlled rough paths to the product {a(u)partial_1^2u}, which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation {partial_t u - P(apartial_1^2 u +sigma f)=0} with rough but given coefficient fields a and {sigma} and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory.