When describing a system of two spin 1/2 particles it is common to separate between a singlet state (with spin zero):
$$ |\psi_0\rangle = \frac{1}{\sqrt2} \left( |+-\rangle - |-+\rangle \right) $$
And a triplet state (with spin 1):
$$ |\psi_1 \rangle = \frac{1}{\sqrt2} \left( |+-\rangle + |-+\rangle \right)$$ $$ |\psi_2 \rangle= |++\rangle $$ $$ |\psi_3 \rangle = |--\rangle $$
My teacher of undergraduate nuclear physics refers to the singlet state as having antiparallel spins, and to the triplet state as having parallel spins. However, I don't understand why the state $|\psi_1\rangle$ is thought of as having parallel spins, since clearly it is a linear combination of states with different spins.
I do understand that the behaviour of the $|\psi_1\rangle$ and $|\psi_0\rangle$ is different under rotations in the spin state, so I guess that my confusion arises from the relative phase of the two states.
Another guess is that I do not understand correctly the addition of angular momentums (spins in this case). The fact that we can separate between a singlet and triplet states is because the Hilbert space is a direct sum of two Hilbert spaces (with dimensions 1 and 3, respectively). Is this only a mathematical fact or does it have a physical interpretation?
Thank you in advance and sorry for the long question