Write the parabola in vertex form: y = a(x - h)^2 + k. For y = x^2 we have y = 1(x - 0)^2 + 0, so a = 1, h = 0, k = 0.
The vertex is (h, k) = (0, 0). The axis of symmetry of a parabola in this form is the vertical line x = h. Therefore the axis of symmetry is x = 0.
Quick check: points (1,1) and (−1,1) are symmetric with respect to the line x = 0, confirming the result.