c = Curve[2 cos(t), 2 sin(t), t, 0, 2 pi]
cos(x + iy) = cos x cos iy - sinx sin iy
= cosh y cos x - i sinh y sin x

parallelX = Curve[cosh(s) * cos(t), - sinh(s) * sin(t), t, 0, 2 pi]
parallelY = Curve[cosh(t) * cos(u), - sinh(t) * sin(u), t, -10, 10]