Step 1: Know the parallel-lines condition.
A general second degree equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ gives a pair of parallel lines only when the top part is a perfect square, which needs $h^2=ab$.
Step 2: Add the second parallel condition.
For the lines to be parallel (not just any pair), the coefficients line up as
\[ \frac{a}{h}=\frac{h}{b}=\frac{g}{f}. \]
Step 3: Pick the useful part.
From $\dfrac{a}{h}=\dfrac{g}{f}$ we cross multiply to get a clean link between the letters.
Step 4: Cross multiply.
\[ af=hg. \]
Step 5: Square both sides.
\[ (af)^2=(hg)^2\ \Rightarrow\ a^2f^2=h^2g^2. \]
Step 6: Read off the answer.
So $g^2h^2=a^2f^2$.
\[ \boxed{g^2h^2=a^2f^2} \]