I originally gave an answer that relied on a claim in Casselman - Introduction to the theory of admissible representations of $p$-adic reductive groups that, as you pointed out by linking to Errata for Casselman's unpublished notes, is wrong; so I deleted the answer. Here's another try.

I hope that I may use $\Theta$ instead of $\theta$ for a subset of the set of simple roots.

The natural map $(w^{-1}P_\Theta w \cap P_\Omega)\backslash P_\Omega \to P_\Theta\backslash P_\Theta w P_\Omega$ is an isomorphism of varieties, so $P_\Theta\backslash P_\Theta w P_\Omega$ is a quasi-projective variety. That seems like it must not be what you were asking for; if not, please let me know, so that I can try to fix this part of the answer.

Since $w^{-1}P_\Theta w \cap P_\Omega$ is smooth, we have that $P_\Omega(k^\text{sep}) \to ((w^{-1}P_\Theta w \cap P_\Omega)\backslash P_\Omega)(k^\text{sep})$, and hence $P_\Omega(k^\text{sep}) \to (P_\Theta\backslash P_\Theta w P_\Omega)(k^\text{sep})$, is surjective.

Since $(w^{-1}P_\Theta w \cap P_\Omega)U_\Omega$ is a pseudo-parabolic subgroup of $P_\Omega$, in the sense of Conrad, Gabber, and Prasad - Pseudo-reductive groups, 2nd edition, Definition 2.2.1, we have that $P_\Omega(k) \to ((w^{-1}P_\Theta w \cap P_\Omega)U_\Omega\backslash P_\Omega)(k)$ is surjective by Lemma C.2.1 there. Let $g$ be a point of $(P_\Theta w P_\Omega)(k^\text{sep})$ whose image in $(P_\Theta\backslash P_\Theta w P_\Omega)(k^\text{sep})$ is ($k$-)rational. Then there is some $h \in P_\Omega(k)$ such that $g$ belongs to $(P_\Theta w U_\Omega h)(k^\text{sep}) = (P_\Theta w U_\Omega)(k^\text{sep})h$. (If you're uncomfortable with pseudo-parabolic subgroups, then it's just a bit of fiddling to handle the genuine parabolic subgroup $w^{-1}P_\Theta w \cap M_\Omega$ of $M_\Omega$ instead.)

It remains to show that $U_\Omega(k) \to (P_\Theta\backslash P_\Theta w U_\Omega)(k)$ is surjective. This is where we come to the claim of Casselman that isn't true for the full quotient $P_\Theta\backslash P_\Theta w P_\Omega$—but we've got down to, as it were, ‘the unipotent part’, where it's fine! We have that $P_\Theta\backslash P_\Theta w U_\Omega$ is isomorphic, as a variety, to $(w^{-1}P_\Theta w \cap U_\Omega)\backslash U_\Omega$. Since both $w^{-1}P_\Theta w \cap U_\Omega$ and $U_\Omega$ are smooth, connected, unipotent subgroups of $G$ that are normalized by $M_\emptyset = C_G(A_\emptyset)$, and since the weights of $A_\emptyset$ on $w^{-1}P_\Theta w \cap U_\Omega$, and on $U_\Omega$, form closed subsets of the relative root system of $G$ with respect to $A_\emptyset$, each is directly spanned by relative root groups (Borel - Linear algebraic groups, Proposition 21.9(ii)). In particular, since both sets of roots are divisible (in the sense that, if $a$ is a relative root such that $2a$ is a weight of $A_\emptyset$ on either group, then $a$ is also a weight of $A_\emptyset$ on that group), we have that there is a section, *as varieties*, of $U_\Omega \to (w^{-1}P_\Theta w \cap U_\Omega)\backslash U_\Omega$, hence of $U_\Omega \to P_\Theta\backslash P_\Theta w U_\Omega$.

All together, we have shown that $P_\Omega(k) \to (P_\Theta\backslash P_\Theta w P_\Omega)(k)$ is surjective, so that we may identity $(P_\Theta\backslash P_\Theta w P_\Omega)(k)$ with $P_\Theta(k)\backslash P_\Theta(k)w P_\Omega(k)$, as you desired.

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