There is the following example when the remark of Borel and Tits is obviously true. Consider the group $G=SL(2, \Bbb D)$ over division algebra $\Bbb D$ for example $\Bbb H$ for $k=\Bbb R$. The parabolic subgroup is the following
\begin{pmatrix} \Bbb H^\times & \Bbb H \\ 0 & \Bbb H^\times \end{pmatrix}
Then it has split rank $1$, so the Bruhat decomposition for $k$-points is just $G_k=P_k\cup P_ksP_k$, where (where s\in N_G(S)$s\in N_G(S)(k)$ (k) ofof course over $\Bbb C$ the cell $P/P$ is the point and the cell $PsP/P$ is open). Hovewer if we pass to the points of $G$ over $\Bbb C$ then $G/P(\Bbb C)$ is the Grassmanian $G(2,4)$ of $2$-dim vector subspaces of $4$-dim. Then there is a $P$-invariant open subset of the unique divisor which consists of $2$-dimensional vector spaces intersecting a given one that is stabilized by $P$. (This subspace correspond to the first block in the matrix and is clear that the condition on the intersection is preserved by the action of $P$). It is clear that being unique such divisor is Galois invariant, however it does not have the points defined over $k=\Bbb R$ on the dense open $P$-orbit.