For y^2=4ax, a point (x1,y1) is inside the parabola if y1^2-4ax1 < 0, on it if y1^2-4ax1 = 0, and outside if y1^2-4ax1 > 0. Note: "inside" the parabola means the region containing the focus. To find the parameter t of a point on the parabola: from y = 2at, we get t = y/(2a), then verify x = at^2.