Parallel / antiparallel spins for two unpaired electrons

The terminology "parallel" and "antiparallel" is a bit loose, whereas the mathematical expressions for the states are exact. Rather than "parallel" and "antiparallel" it might be better to say "with total spin 1" and "with total spin zero", but that is a bit of a mouthful.

You are correct that the terminology does not apply straightforwardly to the state $\frac{1}{\sqrt{2}} (|↓↑\rangle +|↑↓\rangle)$. However, it is significant that that state does have total spin 1, therefore the two spins are in some sense adding up in a parallel as opposed to antiparallel fashion. If you express this state in terms of eigenstates of the x component of spin then you get $$ \frac{1}{\sqrt{2}} (|↓↑\rangle +|↑↓\rangle ) = \frac{1}{2\sqrt{2}} (R-L)(R+L) + \frac{1}{2\sqrt{2}}(R+L)(R-L) \\ = \frac{1}{2\sqrt{2}}\left( RR + RL - LR - LL + RR -RL + LR - LL \right) \\ = \frac{1}{\sqrt{2}} (RR + LL) $$ where I used the notation $$ R = \frac{1}{\sqrt{2}}( |↑\rangle +|↓\rangle ), \;\;\;\; L = \frac{1}{\sqrt{2}}( |↑\rangle -|↓\rangle ) $$ which are "right" and "left" in the sense of eigenstates of x component of spin. So in this basis the state does look a bit more like the spins are parallel to one another.

You could also explore what the other states look like in this basis. It will be instructive.

You cannot identify parallel spins with the triplet and antiparallel spins with the singlet.

The addition of two spins is a special case of the the addition of angular momenta.

In this special case you end up with four states.

Three of the states -- ($|\uparrow\uparrow\rangle$, $|\downarrow\downarrow\rangle$, and $(1/\sqrt{2})(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle$ -- have the same quantum number $s=1$.

The singlet is $(1/\sqrt{2})(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$ and has $s=0$.

This set of lecture notes explains more.