Step-by-step Solution:

1. Identify the vertex of the parabola. The vertex of the parabola \(y^{2} = 4ax\) is at the origin \((0, 0)\). 2. Determine the coordinates of the other two vertices. Let the other two vertices of the equilateral triangle be \((x, y)\) and \((x, -y)\) due to symmetry. 3. Use the equilateral triangle property. The distance between \((0, 0)\) and \((x, y)\) should be equal to the distance between \((x, y)\) and \((x, -y)\): \[ \sqrt{x^{2} + y^{2}} = 2y \] Squaring both sides: \[ x^{2} + y^{2} = 4y^{2} \] Simplifying: \[ x^{2} = 3y^{2} \] 4. Substitute into the parabola equation. Since \((x, y)\) lies on the parabola: \[ y^{2} = 4ax \] Substitute \(x^{2} = 3y^{2}\) into the equation: \[ y^{2} = 4a \sqrt{3}y \] Solving for \(y\): \[ y = 4a \sqrt{3} \] 5. Calculate the side length. The side length \(s\) of the equilateral triangle is the distance between \((0, 0)\) and \((x, y)\): \[ s = \sqrt{x^{2} + y^{2}} = \sqrt{3y^{2} + y^{2}} = \sqrt{4y^{2}} = 2y \] Substituting \(y = 4a \sqrt{3}\): \[ s = 2 \times 4a \sqrt{3} = 8a \sqrt{3} \] Therefore, the length of the side of the equilateral triangle is: \[ \boxed{8a\sqrt{3}} \]