The area of a parallelogram is given by the magnitude of the cross product of its adjacent sides: $\text{Area} = |\vec{P} \times \vec{Q}|$. Computing the cross product: $\vec{P} \times \vec{Q} = (2\hat{i} + 3\hat{j}) \times (\hat{i} + 4\hat{j}) = 8(\hat{i} \times \hat{j}) + 3(\hat{j} \times \hat{i}) = 8\hat{k} - 3\hat{k} = 5\hat{k}$. The magnitude of $5\hat{k}$ is $5$. Hence, the area is $5$ units. Other options represent computational errors in evaluating the cross product magnitude.