Complex line integral along a parabola

Question

I'm trying to evaluate $\int_C (x^2-iy^2)dz$ along the parabola $y=2x^2$ from (1,2) to (2,8). I know the answer is $\frac{511}{3}-\frac{49}{5}i$. I'm pretty sure that I need to parameterize it to make this work. I set $x=t, y=2t^2$. Then I have, $$\int_1^2 (t^4-i4t^4)dt=\frac{1}{3}t^3-i\frac{4}{5}t^5\Big|_1^2=\left(\frac{8}{3}-i\frac{128}{5}\right)-\left(\frac{1}{3}-i\frac{4}{5}\right)=\frac{7}{3}-i\frac{124}{5}$$ I suspect that integrating from $t=1$ to $t=2$ is the problem. What can I do to fix this?