Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position relative to $P$, so that $G = PK$. Under what circumstances do we have $P \cap K = (M \cap K)(N \cap K)$?

My motivation is the following: it is a common convention, e.g. in Harish-Chandra's notes, to normalize a left or right Haar measure on each closed subgroup $H$ of $G$ so that $H \cap K$ has measure one. Supposing we do this for $M$ and $N$, then we get a left Haar measure $d_lp$ on $P$ satisfying:

$$\int\limits_P f(p)d_lp = \int\limits_M \int\limits_N f(mn) dndm$$ My question is then equivalent to the assertion that the Haar measure on $P$ in this way follows the convention that $P \cap K$ has measure one (taking $f = \operatorname{Char} P \cap K$).