The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm,

Parallelogram $A B C D$ is made up of congruent $\triangle A B C$ and $\triangle A D C$.

Area of triangle $A B C=\sqrt{s(s-a)(s-b)(s-c)}$ (Here, $s$ is the semiperimeter.)

Thus, we have:

$s=\frac{a+b+c}{2}$

$s=\frac{34+20+42}{2}$

$s=48 \mathrm{~cm}$

Area of $\Delta A B C=\sqrt{48(48-34)(48-20)(48-42)}$

$=\sqrt{48 \times 14 \times 28 \times 6}$

$=336 \mathrm{~cm}^{2}$

Now,

Area of the parallelogram $=2 \times$ Area of $\Delta A B C$

$=2 \times 336$

$=672 \mathrm{~cm}^{2}$