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Feb 3, 2020 at 14:54 comment added user267839 The singlet corresponds to $S = 0$; which means that the two spins $S_1$ and $S_2$ are antiparallel. The triplet corresponds to an angle $\alpha = 70 deg$ between the two vectors $S_1$ and $S_2$. This is the closest the two spins come to being parallel." That is I think that it might be better to say "with total spin 1" and "with total spin zero" instead of "parallel" and "antiparallel" as Andrew stated. while "parallel" and "antiparallel" in this context seems be done by somebody who prefered to sleep during quantum mechanic lecture :)
Feb 3, 2020 at 14:50 comment added user267839 "You cannot identify parallel spins with the triplet and antiparallel spins with the singlet." This is exactly the intuition I have and which gave rise for this question after facing the notations in the linked text above. It seems that this confusing identifications "parallel spins with the triplet and antiparallel spins with the singlet" are "motivated" by the comment (iv) on page from your linked notes: it says:
Feb 3, 2020 at 8:44 history edited user249968 CC BY-SA 4.0
Replace > with \rangle
Feb 3, 2020 at 4:50 history answered Tony CC BY-SA 4.0