Try Geometry!

Solution using complex numbers:

Let one side of the parallelogram be represented by $a$ ( $a \in \mathbb{C}$ ).

Then the other side is $are^{i\frac{\pi}{3}}$ ( $r \in \mathbb{R}^+$ ) where $r$ is the ratio of the sides.

The given data is equivalent to the following equation: $\frac{|a(1+re^{i\frac{\pi}{3}}))|^2}{|a(1-re^{i\frac{\pi}{3}}))|^2} = 3$

Solving this equation, we get $\boxed{r=1}$

A parallelogram whose acute angle is $60^{\circ}$ is a Rhombus.