I hope you will permit me to write $P^+$ in place of $P$. Put $U^\pm = R_u(P^\pm)$.

Yes, at least in the split case.

Suppose first that $P^+$ and $P^-$ are minimal. Put $T = P \cap P^-$. By working in $\operatorname{SL}_2(k)$ or $\operatorname{PGL}_2(k)$, you see that, for all $t \in k^\times$ and all roots $\alpha$ of $T$ in $G$, you can write $\alpha^\vee(t)$ as a product of $6$ elements of $U^+(k)$ and $U^-(k)$. This covers $T(k)$, using $6r$ elements, where $r$ is the semisimple rank of the group. Once you have written, for each element $w$ of the (finite!) Weyl group of $T$ in $G$, a representative of $w$ as a product of elements of $U^+(k)$ and $U^-(k)$ (which, again by a rank-$1$ computation, can be done using at most $3\ell$ elements, where $\ell$ is the length of a minimal expression for $w$ as a product of reflections), you only need $2$ more elements to generate the corresponding Bruhat cell.

Now continue to suppose that $G$ is split, but drop the assumption that $P^+$ and $P^-$ are minimal. Put $M = P^+ \cap P^-$, and let $T$ be a split maximal torus in $M$.

Let $\alpha$ be a root of $T$ in $M$. Since we (should) have assumed that $P^+$ does not contain any isotropic factor of $G$, there is some root $\beta$ of $T$ in $U^+$ such that $\alpha + \beta$ is also a root of $T$ in $U^+$. Then, for a suitable Chevalley–Steinberg system $(u_r : \operatorname{Add} \to U_r)_{\text{$r$ a root}}$, we have for all $t \in k$ that $[u_{-\beta}(t), u_{\alpha + \beta}(1)]$ lies in $u_\alpha(t)U^-(k)$. That is, each element of $U_\alpha(k)$ is a product of $4$ elements of $U_{-\beta}(k) \subseteq U^-(k)$ and $U_{\alpha + \beta}(k) \subseteq U^+(k)$ with an element of $U^-(k)$.

This shows that the group of $k$-rational points of every root subgroup of $M$ is boundedly generated by $U^+(k)$ and $U^-(k)$; so, if $B_M^\pm$ are opposite Borel subgroups of $M$ containing $T$, then $R_u(B_M^\pm)(k)$ are boundedly generated by $U^+(k)$ and $U^-(k)$; so $R_u(B_M^\pm)(k)U^\pm(k) = R_u(B_M^\pm\cdot U^\pm)(k)$ is boundedly generated by $U^+(k)$ and $U^-(k)$. Since $B_M^+\cdot U^+$ and $B_M^-\cdot U^-$ are opposite Borel subgroups of $G$, we have reduced to the previous case.

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