We consider optimization problems formulated in the tropical mathematics setting to minimize two competing objective functions defined for vectors over an idempotent semifield, subject to linear inequality and box constraints on the decision vector. The objective functions may involve (multiplicative) conjugate transposition of vectors and take the form of pseudo-linear and pseudo-quadratic functions. To derive all Pareto-optimal solutions of a problem, we apply an approach that introduces two parameters to represent the optimal values of the objectives in the Pareto frontier. The constrained bi-objective problem under solution then reduces to a system of parametrized vector inequalities. We proceed in two steps: first, we use the existence conditions for solutions of the system to evaluate the parameters, and second, we take all solutions of the system, which correspond to the parameters, as the Pareto-optimal solution to the initial bi-objective problem. As an application of the approach, we present solutions of optimization problems that appear in project scheduling, decision making and location analysis.