We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BNpair of $G$. How we prove that every parabolic subgroup of $G$ containing $B$ is of the form $P=BW_{T}B$, where $W$ is the Weyl group of the BNpair $(B,N)$, it's the same as a Weyl group of $G$, and $W_{T}=<T>$ a standard subgroup of $W$ ($W$ is a Coxeter group with set of generators $S$, and $T$ is a subset of $S$ and finally $W_{T}$ is a subgroup generated by $T$).
Combine Theorem 6.43 (1) pg. 315 in AbramenkoBrown "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) 


To provide a more balanced context for the question (and the answer by pm), it's useful to separate the elementary notion of BNpair from the far more sophisticated structure theory of algebraic groups. Following Chevalley's 1955 Tohoku paper on finite simple groups of Lie type and his 195658 Paris seminar on classification of semisimple algebraic groups, Jacques Tits formulated an axiomatic framework to cover these and further examples. In a concise 1962 Comptes Rendus note, he defined BNpairs and derived their main properties including the role of "Borel" and "parabolic" subgroups involved in answering Rajkarov's question. Chapter 4 of the Bourbaki treatise Groupes et algebres de Lie (1968) adopted the name "systemes de Tits" for BNpairs and included in the exercises much of the contemporary development by Tits of the theory of buildings. By now there are many treatments of BNpairs in books, papers, and lecture notes, including the selfcontained Section 29 of my 1975 Springer text on linear algebraic groups. But the theory of buildings lies beyond this elementary axiomatic stage and is not directly relevant to the question asked here. Similarly, the theory of BNpairs is independent of algebraic groups but flexible enough to fit more advanced developments such as the IwahoriMatsumoto study of groups over $p$adic fields. (The latter IHES paper is now freely available online, but I'm not sure whether any other online sources contain complete details about BNpairs.) Maybe it's also worth mentioning that quite a few MO questions involve BNpairs, also known as (B,N)pairs or Tits systems, as well as buildings. So it may be worthwhile to browse through these. 

