Given that $O$ is the centroid of $\triangle ABC \Rightarrow \frac{AO}{OD}=2$.
Applying Menelaus' theorem in $\triangle POD$, we get
$$
\begin{array}{l}
\frac{OA}{AD} \cdot \frac{DN}{NP} \cdot \frac{PM}{MO}=1 \Rightarrow \frac{DN}{NP}=\frac{3}{2} \\
\Rightarrow \frac{PN}{PD}=\frac{2}{5} \Rightarrow \frac{S_{\triangle PEF}}{S_{\text {trapezoid EFCB }}}=\frac{4}{21} .
\end{array}
$$

Therefore, the volume ratio of the upper and lower parts of the tetrahedron is $\frac{4}{21}$.
13. Let the height of the regular tetrahedron $P-ABC$ be $PO$, and let $M$ be the midpoint of $PO$. A plane passing through $AM$ and parallel to the edge $BC$ divides the tetrahedron into upper and lower parts. Find the ratio of the volumes of these two parts.