It is difficult to reconcile your first two statements, because they are actually wrong as written!
A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his classification of real semisimple Lie algebras. My favourite reference for this is Besse's Einstein manifolds, Chapter 7F.
I suspect the source of the confusion might be with the meaning of "constant curvature", which usually means constant sectional curvature (or perhaps parallel Riemann curvature) and not constant scalar curvature, which is a much weaker condition.
Concerning your final parenthetic question, there is a theorem of Ambrose and Singer, reformulated by Kostant, which says that a riemannian manifold $(M,g)$ is locally homogeneous if and only if it admits a metric connection with parallel torsion and parallel curvature. It is locally symmetric if (and only if) the connection is torsion-free.