We introduce a censored version of the stochastic six-vertex model. We show that for parameters $b_1 < b_2$, this model started from the initial condition ${1}_{x>0}$ is stochastically dominated at any time by the blocking measure. This is a partial analog of the censoring inequality for monotone spin systems. In particular, this result allows us to control the behavior of second-class particles. The proof uses parabolic Kazhdan--Lusztig $R$-polynomials, whose appearance is explained using a connection between the stochastic six-vertex model and the Iwahori--Hecke algebras of symmetric groups. Furthermore, we find an intertwining relation for this process using normalized parabolic Kazhdan--Lusztig $R$-polynomials as an intertwining kernel.