Parallel covering of a square with equilateral triangles

Suppose that $I$ is a unit square. Let $\triangle$ be an equilateral triangle with a side parallel to a side of $I$, and let $\{\triangle_{n}\}$ be a collection of the homothetic copies of $\triangle$. In this note, we show that if the total sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $1+\frac{7\sqrt{3}}{12}$, then these equilateral triangles can parallel cover the unit square $I$, and this bound is optimal.