For a surface in $\mathbb{R}^3$, if $(x,y,z) = \sigma(u,v)$ is one of its coordinate patches, the regularity requirement says that the partial derivative vectors $\frac{\partial \sigma}{\partial u}$ and $\frac{\partial \sigma}{\partial v}$ must be linearly independent.
The examples I've seen where a point on a surface "looks" irregular, like the apex of a cone, or the origin for the surface $(u^3, v^3, uv)$, all have at least one of these partial derivatives vanishing. Is there an accessible example in which both partials are nonzero, but they're parallel?
One most probably can come up with a much easier example. But as soon as I read your question this example came to my mind:
Let $N: \mathbb{R}^2 \rightarrow\mathbb{S}^2$ be the following parametrization: $$ N\,(u,v) = \left(\begin{array}{c} \tanh{(u + v)}\,\cos{({u-v})}\\ \tanh{(u + v)}\,\,\sin{({u-v})}\\ \frac{2\,(\cosh{(u+v)} + \sinh{(u+v)})}{1 + \cosh{(2u+2v)} + \sinh{(2u+2v)}} \end{array}\right), $$
At $(u,v) = (0,0)$ the tangents exist but both are parallel: $N_u (0,0) = N_v(0,0) = (1,0,0)$. The above parametrization above is sometimes referred to as the trivial Chebyshev net on the sphere (more accurately, it is just one of the members of a family of such nets).
Below you can find a Maple plot of it with the red arrow standing as both $N_u$ and $N_v$ at the point $N(0,0)$: