This is about the addition of angular momenta and not about "parametric downconversion", which is way more complicated. The mathematical question is that after the eigenvalue problem of the total-spin operator which is a sum of two independent spins. You have to get familiar with the tensor product of Hilbert spaces (in this case the somewhat simpler case of finite-dimensional Hilbert spaces) and how to solve the eigenvalue problem of angular-momentum operators (in this case simplifying to the eigenvalue problem of finite-dimensional hermitean matrices).
As will turn out in your case, adding two spins 1/2, the sum contains vectors of total spin 0 (singlet) and total spin 1 (triplet), and the basis of eigenvectors is
for total spin 1:
$$|S=1,\Sigma_z=1 \rangle=|s_1=1/2,\sigma_z^{(1)}=1/2;s_2=1/2,\sigma_z^{(2)}=1/2 \rangle,$$
$$|S=1,\Sigma_z=0 \rangle=\frac{1}{\sqrt{2}} (|s_1=1/2,\sigma_z^{(1)}=1/2;s_2=1/2,\sigma_z^{(2)}=-1/2 \rangle + |s_1=1/2,\sigma_z^{(1)}=-1/2;s_2=1/2,\sigma_z^{(2)}=1/2 \rangle),$$
$$|S=1,\Sigma=-1 \rangle = |s_1=1/2,\sigma_z^{(1)}=-1/2;s_2=1/2,\sigma_z^{(2)=-1/2}.$$
for total spin 0:
$$|S=0,\Sigma_z=0 \rangle = \frac{1}{\sqrt{2}} (|s_1=1/2,\sigma_z^{(1)}=1/2;s_2=1/2,\sigma_z^{(2)}=-1/2 \rangle - |s_1=1/2,\sigma_z^{(1)}=-1/2;s_2=1/2,\sigma_z^{(2)}=1/2 \rangle).$$