The author considers the weighted Triebel spaces of product type related to the case when \(R^ n\) carries a structure \(R^ n=R_ 1^ n\times\dots\times R_ N^ n\) where each \(R_ 1^ n\) is provided with a distance function \(\rho_ i\) which is called parabolic. It is generated by a dilation group \(p_ t:=\exp(Q\ln t)\) with a real \(n\times n\) matrix \(Q\) (as in the known papers by Calderon and Torchinsky, Stein and Wainger, Dappa and others). Instead of a maximal technique the author uses a pointwise estimate developing his earlier results in the isotropic case. The author obtains sharp versions of Fourier multipliers theorems on weighted and unweighted Triebel spaces, \(L^ p\)-spaces and vector- valued \(L^ p\)-spaces. He also studies some generalizations of the Hörmander class \(S_{1,\delta}^ m\), \(0\leq \delta\leq 1\), gives an estimation of pseudo-differential operators \((PDOs)\) on the basic of the same pointwise estimate and considers \(PDOs\) with symbols in the Besov space \(B_{\infty,q}^ r\) in the \(x\)-variable on weighted Triebel spaces of product type.