It's easy to recognize the points of a graph of a continuous functions in $\mathbb R^2$ which have zero derivative. They are points where the tangent is parallel to the horizontal axis. So in everyday usage and daily exercises in calculus classes (there are exceptions for example $y=x^3$) these graphs has peaks like the origin in graph of $y=x^2$.
So I would like to know how to recognize the points where a parametrized curve $\alpha:[a,b]\to \mathbb R^2$ has zero derivative.
The main point about parametrizations is that a curve viewed as a subset of $\mathbb R^2$ admits many different paramatrizations. So if you want to spot vanishing of the derivative, you have to watch a point moving through $\mathbb R^2$ and not just see its path afterwards. Seeing the point move, the derivative is just the velocity in a point, so vanishing of the derivative means that the velocity is zero at a given time.
If you are interested in point at which there cannot be any regular parametrization, then you have to look for cusps or corners as suggested in the comment by @TedShifrin.