= Solution
For both modes, a leading weak quark transition in $K^+=u\bar s$ is $\bar s\to\bar u+W^+$, followed by $W^+\to u\bar d$, with the original $u$ acting as a <spectator quark>. This produces the four quarks needed for $\pi^+\pi^0$, using the $u\bar u$ component of $\pi^0$. Producing three pions additionally requires a $d\bar d$ pair from strong <hadronization>. A representative quark-line diagram draws it as gluon emission and splitting.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2021-kaon-decay-diagrams.png]
{title=Representative weak quark diagrams for the two-pion and three-pion kaon decays}
The three-pion assignment is $u\bar d$, $u\bar d$, and $d\bar u$, matching $\pi^+\pi^+\pi^-$. The extra soft pair creation is a schematic representation of <hadronization>, not a controlled perturbative prediction of the relative decay rates. Both processes are first order in the effective weak interaction.
There is an explicit error or hypothetical assumption to address: the original PDF assigns \b[$0^+$ to all these mesons]. Taken literally, the two-pion final state has $L=0$, because all particles have zero <spin> and the initial $J$ is zero. Its <parity> is $(+1)^2(-1)^0=+1$. For three spinless particles, use two Jacobi <orbital angular momentum> quantum numbers $l,L$. Coupling to total zero requires $l=L$, so the orbital parity $(-1)^{l+L}$ is $+1$ and the three-particle parity is also $+1$. Thus \b[under the printed $0^+$ assignment, both channels conserve parity].
The named physical <pions> and <kaons> are instead <pseudoscalar mesons> with $J^P=0^-$. With those physical assignments,
$$
P_{2\pi}=(-1)^2(-1)^0=+1,\qquad
P_{3\pi}=(-1)^3(-1)^{l+L}=-1,
\qquad P_K=-1.
$$
Therefore
$$
\boxed{\text{Physical mesons: }K^+\to2\pi\text{ violates parity; }K^+\to3\pi\text{ preserves parity}.}
$$
These two opposite-parity final states from the same physical kaon are a classic illustration of <parity violation> in a weak decay. The physical conclusion requires correcting the printed parity assignment explicitly.
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