  Parabolic $R$-polynomials were introduced by Deodhar as parabolic analogues
of ordinary $R$-polynomials defined by Kazhdan and Lusztig. In this paper, we
are concerned with the computation of parabolic $R$-polynomials for the
symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$, and
let $S=\{s_i\,|\, 1\leq i\leq n-1\}$ be the generating set of $S_n$, where for
$1\leq i\leq n-1$, $s_i$ is the adjacent transposition. For a subset
$J\subseteq S$, let $(S_n)_J$ be the parabolic subgroup generated by $J$, and
let $(S_n)^{J}$ be the set of minimal coset representatives for $S_n/(S_n)_J$.
For $u\leq v\in (S_n)^J$ in the Bruhat order and $x\in \{q,-1\}$, let
$R_{u,v}^{J,x}(q)$ denote the parabolic $R$-polynomial indexed by $u$ and $v$.
Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S\setminus\{s_i\}$, and
obtained an expression for $R_{u,v}^{J,x}(q)$ when
$J=S\setminus\{s_{i-1},s_i\}$. We introduce a statistic on pairs of
permutations in $(S_n)^J$ for $J=S\setminus\{s_{i-2},s_{i-1},s_i\}$. Then we
give a formula for $R_{u,v}^{J,x}(q)$, where
$J=S\setminus\{s_{i-2},s_{i-1},s_i\}$ and $i$ appears after $i-1$ in $v$. We
also pose a conjecture for $R_{u,v}^{J,x}(q)$, where
$J=S\setminus\{s_{k},s_{k+1},\ldots,s_i\}$ with $1\leq k\leq i\leq n-1$ and the
elements $k+1,k+2,\ldots, i$ appear in increasing order in $v$.
