A General Statement I read is:
Given a parametric curve $$\vec r(s)=(x(s),y(s))$$ the tangent vector is represented by the first derivative, $$\vec r'(s)=(x'(s),y'(s))$$ Then the vector $\vec n=\vec r''(s)$ is perpendicular to the Tangent and directed towards the Center of curvature. Divide it by its modulus and you have the principal normal vector to the curve.
This Statement is not valid if the parametric curve is a second order polynomial one. In this case the second derivative of the $\vec r(s)$ vector is a constant vector, and it cannot be normal to the curve everywhere. Nevertheless the curve has a principal normal vector and a Center of curvature (were it a first order polynomial, the Center of curvature would be at infinity but for a second order polynomial the Center of curvature is well defined).
The initial Statement seems not to be that general. What is its Domain of validity?
Note the use of "$s$" as the parameter. $s$ traditionally denotes arc-length along the curve. I would bet that somewhere preceding this statement, this restriction was mentioned: they are only talking about arclength parametrizations.
Because the formula for arclength (for an arbitrary parametrization by $t$) is $\left(\frac{ds}{dt}\right)^2 = \|\dot{\vec r}\|^2$, when $t = s$, we have $$\vec r' \cdot \vec r' = \|\vec r'\|^2 = 1$$ When you differentiate that formula by $s$ again, you get $$\vec r'' \cdot \vec r' + \vec r' \cdot \vec r'' = 0$$ and therefore just $$\vec r'' \cdot \vec r' = 0$$ I.e., when the curve is parametrized by arc length, the second derivative of $\vec r$ is indeed always normal to the first.