Jacob,
Can you re-run Mathematica using the following parametric equations instead?
Code:
x_lower = t + 2*a*t / sqrt(1 + (2*a*t)^2);
y_lower = a*t^2 - 1 / sqrt(1 + (2*a*t)^2);
x_upper = t - 2*a*t / sqrt(1 + (2*a*t)^2);
y_upper = a*t^2 + 1 / sqrt(1 + (2*a*t)^2);