diff --git a/src/constrained_variables.jl b/src/constrained_variables.jl
index 681a250..89c1f31 100644
--- a/src/constrained_variables.jl
+++ b/src/constrained_variables.jl
@@ -89,10 +89,11 @@ function _get_parameter_variables(::PrimalDualMap{T}, primal_model) where {T}
         primal_model,
         MOI.ListOfConstraintIndices{MOI.VariableIndex,MOI.Parameter{T}}(),
     )
-    parameters = Set{MOI.VariableIndex}()
+    parameters = Dict{MOI.VariableIndex,MOI.Parameter{T}}()
     for ci in cis
         vi = MOI.get(primal_model, MOI.ConstraintFunction(), ci)
-        push!(parameters, vi)
+        set = MOI.get(primal_model, MOI.ConstraintSet(), ci)
+        parameters[vi] = set
     end
     return parameters
 end
diff --git a/src/dual_model_variables.jl b/src/dual_model_variables.jl
index 980fb65..73280be 100644
--- a/src/dual_model_variables.jl
+++ b/src/dual_model_variables.jl
@@ -114,6 +114,7 @@ function _add_primal_parameter_vars(
     primal_model::MOI.ModelLike,
     primal_dual_map::PrimalDualMap{T},
     dual_names::DualNames,
+    moi_parameter_sets::Dict{MOI.VariableIndex,MOI.Parameter{T}},
     variable_parameters::Vector{MOI.VariableIndex},
     primal_objective,
     ignore_objective::Bool,
@@ -134,15 +135,22 @@ function _add_primal_parameter_vars(
         variable_parameters
     end
     vis = MOI.add_variables(dual_model, length(parameters))
+
+    moi_parameters = keys(moi_parameter_sets)
     for i in eachindex(vis)
-        primal_dual_map.primal_parameter_to_dual_parameter[parameters[i]] =
-            vis[i]
+        pvi = parameters[i]
+        dvi = vis[i]
+
+        if pvi in moi_parameters
+            MOI.add_constraint(dual_model, dvi, moi_parameter_sets[pvi])
+        end
+        primal_dual_map.primal_parameter_to_dual_parameter[pvi] = dvi
         if !is_empty(dual_names)
-            vi_name = MOI.get(primal_model, MOI.VariableName(), parameters[i])
+            pvi_name = MOI.get(primal_model, MOI.VariableName(), pvi)
             prefix =
                 dual_names.parameter_name_prefix == "" ? "param_" :
                 dual_names.parameter_name_prefix
-            MOI.set(dual_model, MOI.VariableName(), vis[i], prefix * vi_name)
+            MOI.set(dual_model, MOI.VariableName(), dvi, prefix * pvi_name)
         end
     end
     return
diff --git a/src/dualize.jl b/src/dualize.jl
index f14a5db..d3dcdad 100644
--- a/src/dualize.jl
+++ b/src/dualize.jl
@@ -87,10 +87,10 @@ function dualize(
     supported_constraints(con_types) # Errors if constraint cant be dualized
 
     # merge user listed parameters and MOI.VariableIndex-in-MOI.Parameter{T}
-    moi_parameters =
+    moi_parameter_sets =
         _get_parameter_variables(dual_problem.primal_dual_map, primal_model)
     all_parameters = Set{MOI.VariableIndex}(_variable_parameters)
-    for p in moi_parameters
+    for p in keys(moi_parameter_sets)
         push!(all_parameters, p)
     end
     variable_parameters = collect(all_parameters)
@@ -165,6 +165,7 @@ function dualize(
         primal_model,
         dual_problem.primal_dual_map,
         dual_names,
+        moi_parameter_sets,
         variable_parameters,
         primal_objective,
         ignore_objective,
diff --git a/test/Tests/test_partial_dual_quadratic.jl b/test/Tests/test_partial_dual_quadratic.jl
index a064570..14af7ac 100644
--- a/test/Tests/test_partial_dual_quadratic.jl
+++ b/test/Tests/test_partial_dual_quadratic.jl
@@ -180,6 +180,7 @@
         @test Set(list_of_cons) == Set(
             [
                 (MOI.VariableIndex, MOI.GreaterThan{Float64})
+                (MOI.VariableIndex, MOI.Parameter{Float64})
                 (MOI.ScalarAffineFunction{Float64}, MOI.EqualTo{Float64})
             ],
         )
@@ -288,6 +289,18 @@
         @test primal_parameter_to_dual_parameter[MOI.VariableIndex(3)] ==
               MOI.VariableIndex(2 + 1)
 
+        @test MOI.get(
+            dual_model,
+            MOI.ConstraintSet(),
+            MOI.ConstraintIndex{MOI.VariableIndex,MOI.Parameter{Float64}}(
+                primal_parameter_to_dual_parameter[MOI.VariableIndex(3)].value,
+            ),
+        ) == MOI.get(
+            primal_model,
+            MOI.ConstraintSet(),
+            MOI.ConstraintIndex{MOI.VariableIndex,MOI.Parameter{Float64}}(3),
+        )
+
         primal_var_in_quad_obj_to_dual_slack_var =
             primal_dual_map.primal_var_in_quad_obj_to_dual_slack_var
         @test primal_var_in_quad_obj_to_dual_slack_var[MOI.VariableIndex(1)] ==