diff --git a/test/jump_tests.jl b/test/jump_tests.jl
index d36e3ed6..da64ae35 100644
--- a/test/jump_tests.jl
+++ b/test/jump_tests.jl
@@ -1500,6 +1500,11 @@ function test_jump_psd_cone_without_parameter_v_and_vv()
     @test is_valid(model, con)
     optimize!(model)
     @test value(x) ≈ 0.50000 atol = 1e-5
+    @test MOI.get(
+        backend(model),
+        MOI.ConstraintPrimalStart(),
+        index(ParameterRef(p)),
+    ) == 1.0
     delete(model, con)
     unregister(model, :con)
     @test_throws MOI.UnsupportedConstraint @constraint(
@@ -1510,7 +1515,7 @@ function test_jump_psd_cone_without_parameter_v_and_vv()
     @test_throws MOI.UnsupportedConstraint @constraint(
         model,
         con,
-        [x * x, (x - 1), p] in MOI.PositiveSemidefiniteConeTriangle(2)
+        [x * x, (x - 1), 0.0] in MOI.PositiveSemidefiniteConeTriangle(2)
     )
     @test_throws MOI.UnsupportedConstraint @constraint(
         model,
@@ -1522,6 +1527,19 @@ function test_jump_psd_cone_without_parameter_v_and_vv()
         con,
         [x, p * (x - 1), x * x] in MOI.PositiveSemidefiniteConeTriangle(2)
     )
+    @test_throws ErrorException(
+        "Constraint attribute MathOptInterface.ConstraintName() cannot be set for $(index(ParameterRef(p)))",
+    ) MOI.set(
+        backend(model),
+        MOI.ConstraintName(),
+        index(ParameterRef(p)),
+        "name",
+    )
+    @test_throws MOI.UnsupportedAttribute MOI.supports(
+        backend(model),
+        MOI.ConstraintName(),
+        MOI.ConstraintIndex{MOI.VariableIndex,MOI.Parameter},
+    )
 end
 
 function test_variable_and_constraint_not_registered()
diff --git a/test/moi_tests.jl b/test/moi_tests.jl
index 9d81adfd..ab94eb17 100644
--- a/test/moi_tests.jl
+++ b/test/moi_tests.jl
@@ -1821,6 +1821,11 @@ function test_getters()
         MOI.Zeros,
         POI.ParametricVectorAffineFunction{Float64},
     )
+    T4 = (
+        MOI.VectorAffineFunction{Float64},
+        MOI.Zeros,
+        POI.ParametricVectorQuadraticFunction{Float64},
+    )
     @test MOI.get(optimizer, POI.ListOfParametricConstraintTypesPresent()) ==
           [T1]
     @test length(
@@ -1853,6 +1858,16 @@ function test_getters()
             }(),
         ),
     )
+    @test isempty(
+        MOI.get(
+            optimizer,
+            POI.DictOfParametricConstraintIndicesAndFunctions{
+                T4[1],
+                T4[2],
+                T4[3],
+            }(),
+        ),
+    )
     MOI.add_constraint(optimizer, 2.0x * p + 2.0p, MOI.LessThan(0.0))
     @test MOI.get(optimizer, POI.ListOfParametricConstraintTypesPresent()) ==
           [T1, T2] ||
@@ -1888,6 +1903,16 @@ function test_getters()
             }(),
         ),
     )
+    @test isempty(
+        MOI.get(
+            optimizer,
+            POI.DictOfParametricConstraintIndicesAndFunctions{
+                T4[1],
+                T4[2],
+                T4[3],
+            }(),
+        ),
+    )
     terms = [
         MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),
         MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, p)),
@@ -1924,6 +1949,16 @@ function test_getters()
             }(),
         ),
     ) == 1
+    @test isempty(
+        MOI.get(
+            optimizer,
+            POI.DictOfParametricConstraintIndicesAndFunctions{
+                T4[1],
+                T4[2],
+                T4[3],
+            }(),
+        ),
+    )
     @test MOI.get(optimizer, POI.ParametricObjectiveType()) == Nothing
     MOI.set(
         optimizer,