  We sharpen in this work the tools of paracontrolled calculus in order to
provide a complete analysis of the parabolic Anderson model equation and
Burgers system with multiplicative noise, in a $3$-dimensional Riemannian
setting, in either bounded or unbounded domains. With that aim in mind, we
introduce a pair of intertwined space-time paraproducts on parabolic H\"older
spaces, with good continuity, that happens to be pivotal and provides one of
the building blocks of higher order paracontrolled calculus.
