Combine Theorem 6.43 (1) pg. 315 in Abramenko-Brown "Buildings" together with Proposition 6.36 (6) on pg. 310.
These are two steps: As you mention, $BW_T B$ is a subgroup (Proposition 6.36).
$BW_TB$ contains $B$, any subgroup containing the Borel subgroup is parabolic, and any parabolic is conjugate to a subgroup containing a specific Borel $B$. (Theorem 6.43)