Parabola obstruction to countable line coverings

A parabola parametrized by $(t,t^2)$ is in bijection with $\mathbb{R}$ and is an uncountable set. Every affine line meets it in at most two points. A countable family of affine lines could therefore cover only countably many points of this parabola, and cannot cover the plane. More generally, an uncountable curve with finite intersection with every permitted member obstructs a countable covering by those members.