Direction of spins in the singlet vs. triplet state

Suppose we consider the first excited state of the helium atom. We know that the first excited state of helium can exist as a triplet or singlet. The possible functions related to the spin of the two electrons in the triplet state are


$$\alpha(1)\alpha(2)$$
$$\beta(1)\beta(2)$$
$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) + \beta(1)\alpha(2) ] $$


while the one for the singlet state is


$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) - \beta(1)\alpha(2) ]$$


The triplet state predicts that the spins of the two electrons are parallel, but according to this equation


$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) + \beta(1)\alpha(2) ] $$


there is a 50% probability that electron 1 is in the alpha state and a 50% probability that it is in the beta state: the same goes for electron 2.
So, if this function predicts that the two spins are antiparallel, why is it part of one of the triplet states?


