Parabolic Kazhdan-Lusztig R-polynomials for quasi-minuscule quotients

We give explicit combinatorial formulas for the parabolic Kazhdan-Lusztig R-polynomials of the quasi-minuscule quotients of the classical Weyl groups. As an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary Kazhdan-Lusztig R-polynomials.