Is the derivative of the unit normal vector parallel to the tangent vector?

As Spock once said, you’re exhibiting two-dimensional thinking: $N\perp T$ and $N'\perp N$ doesn’t imply that $T$ and $N'$ are parallel. Consider the standard coordinate axes: the $y$-axis is orthogonal to the $x$-axis and the $z$-axis is orthogonal to the $y$-axis, but the $z$-axis certainly isn’t parallel to the $x$-axis.