In this paper, we consider parametric equilibrium problems with respect to the lexicographic order. Firstly, we introduce a relaxed version of the triangle inequality and apply it to formulate existence conditions of solutions to such problems. Next, we suggest and study many types of well-posedness for the reference problems, as well as analyze in details the relationships between these properties. Then after, based on these relationships, we formulate some sufficient conditions for the mediating concept of the suggested properties. Finally, as an application, we consider optimization problems with optimization constraints and discuss the relationships between these problems and the equilibrium problems with respect to the lexicographic order.