  In this paper, we mainly study left invariant pseudo-Riemannian
Ricci-parallel metrics on connected Lie groups which are not Einstein.
Following a result of Boubel and B\'{e}rard Bergery, there are two typical
types of such metrics, which are characterized by the minimal polynomial of the
Ricci operator. Namely, its form is either $(X-\alpha)(X-\bar{\alpha})$ (type
I), where $\alpha\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II).
Firstly, we obtain a complete description of Ricci-parallel metrics of type I.
In particular, such a Ricci-parallel metric is uniquely determined by an
Einstein metric and an invariant symmetric parallel complex structure up to
isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type
II by using double extension process. Surprisingly, we find that every double
extension of a metric Abelian Lie algebra is Ricci-parallel and the converse
holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II.
Moreover, we construct infinitely many new explicit examples of Ricci-parallel
metric Lie algebras which are not Einstein.
