No Meeting

Some lines miss the parabola entirely. Take $y = {x}^{2}+3$ and $y = 2x$. Substituting gives ${x}^{2}+3 = 2x$, so ${x}^{2}-2x+3 = 0$.

Its discriminant is ${\left(-2\right)}^{2}-4\cdot 1\cdot 3$, which is $4-12$, or -8. Negative, so the quadratic has no real solutions, only a pair of complex ones. A meeting point has to sit on the grid, with real coordinates, so there is none: the system has no solution. The line passes below the parabola without touching it, and the complex roots do not matter here, because the question was about points on the plane.