It is shown that para-multiplication applies to a certain product \pi(u,v) defined for appropriate temperate distributions u and v. Boundedness of \pi(\cdot,\cdot) is investigated for the anisotropic Besov and Triebel--Lizorkin spaces, more precisely for B^{M,s}_{p,q} and F^{M,s}_{p,q} with s\in\mathbb{R} and p and q\in\,]0,\infty], though p<\infty in the F-case. Both generic as well as various borderline cases are treated. The spaces B^{M,s_0}_{p_0,q_0}\oplus B^{M,s_1}_{p_1,q_1} and F^{M,s_0}_{p_0,q_0}\oplus F^{M,s_1}_{p_1,q_1} to which \pi(\cdot,\cdot) applies are determined in the case \max(s_0,s_1)>0. For generic isotropic spaces F^{s_0}_{p_0,q_0}\oplus F^{s_1}_{p_1,q_1} the receiving F^{s}_{p,q} spaces are characterised. It is proved that \pi(f,g)=f\cdot g holds for functions f and g when f\cdot g is locally integrable, roughly speaking. In addition, \pi(f,u)=fu when f is of polynomial growth and u is temperate. Moreover, for an arbitrary open set \Omega in Euclidean space, a product \pi_\Omega(\cdot,\cdot) is defined by lifting to \mathbb{R}^n. Boundedness of \pi on \mathbb{R}^n is shown to carry over to \pi_\Omega in general.