Define the relationship between the base and height.

Let $h$ be the height of the parallelogram and $b$ be the base.
According to the problem, the base is four times the height:

$$b = 4h$$

The area ($A$) of a parallelogram is given by the formula $A = b \times h$.
We are given that the area is $576 \text{ cm}^2$.
Substitute $b = 4h$ into the area formula:

$$576 \text{ cm}^2 = (4h) \times h$$ $$576 \text{ cm}^2 = 4h^2$$

Divide both sides by 4:

$$h^2 = \frac{576 \text{ cm}^2}{4}$$ $$h^2 = 144 \text{ cm}^2$$

Take the square root of both sides:

$$h = \sqrt{144 \text{ cm}^2}$$ $$h = 12 \text{ cm}$$

The height of the parallelogram is $12 \text{ cm}$.

The final answer is $\text{B) } 12 \text{ cm}$.