Basically the question is in the title but, I've been told to think of affine subspaces are spaces parallel to each other.
So, say we're talking about: $$F_2^2.$$
Wouldn't that mean that only one contains $(0,0)$, because if any other subspace contained $(0,0)$, they wouldn't be parallel?
And I thought a subspace needed to contain $(0,0)$ (if we define addition of $(a, b) + (c d) = (ac, bd)$), so wouldn't that mean that only one affine subspace is a linear subspace? Thanks!
No, not in general. In the usual plane, the line $L$ through $(-1,0)$ and $(0,1)$ is an affine subspace within the real vector space $\mathbb{R}^2$. A (proper) linear subspace needs to contain the origin.
Note $L$ is parallel to $W$, the span of $(1,1)$. There are many affine subspaces parallel to $W$. Actually all of these can be written $(0,t) + W$ as $t$ runs through all reals.
Only one contains the origin: the one such that $t=0$.
In this example, only when $t=0$ is $(0,t) + W$ a (proper) linear subspace. A linear subspace is a special affine subspace.
Nevertheless, you should be prepared for authors to simply say "subspace". Whether the general affine sense is meant, or if the narrower (linear algebra) sense is meant you might need to judge from the context or the introductory remarks.