*Tangent* at a Point and *Parallel* at a Point

Question

If we say that a curve is tangent to a function at a point, is that equivalent to saying that the curve is parallel to said function at said point?

Answer 1

I'm guessing the difference is that a parallel curve doesn't necessarily intersect the function at said point but a tangent curve does.

For example: the parabola $f(x)=x^2$ and the line $g(x)=0$ are tangent at $x=0$, but $f(x)$ is parallel to the line $h(x)=-1$ at $x=0$.