We consider $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds satisfying certain curvature conditions: $R(\xi,X)\cdot S=0$ and $S\cdot R(\xi,X)=0$. We prove that on a Lorentzian para-Sasakian manifold $(M,\varphi,\xi,\eta,g)$, if the Ricci curvature satisfies one of the previous conditions, the existence of $\eta$-Ricci solitons implies that $(M,g)$ is Einstein manifold. We also conclude that in these cases there is no Ricci soliton on $M$ with the potential vector field $\eta$. On the other way, if $M$ is of constant curvature, then $(M,g)$ is elliptic manifold. Cases when the Ricci tensor satisfies different other conditions are also discussed.