diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index b5302628..a3c8aa63 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -323,6 +323,22 @@ function MOI.supports(
     return MOI.supports(model.optimizer, attr, tp)
 end
 
+function MOI.get(
+    model::Optimizer,
+    attr::MOI.VariablePrimalStart,
+    v::MOI.VariableIndex,
+)
+    if _variable_in_model(model, v)
+        return MOI.get(model.optimizer, attr, v)
+    elseif _parameter_in_model(model, v)
+        # this is effectivelly a no-op, but we do validation
+        _val = model.parameters[p_idx(v)]
+        return _val
+    else
+        error("Variable not in the model")
+    end
+end
+
 function MOI.set(
     model::Optimizer,
     attr::MOI.VariablePrimalStart,
@@ -413,17 +429,25 @@ end
 function MOI.is_valid(
     model::Optimizer,
     c::MOI.ConstraintIndex{F,S},
-) where {
-    F<:Union{MOI.VariableIndex,MOI.VectorOfVariables,MOI.VectorAffineFunction},
-    S<:MOI.AbstractSet,
-}
+) where {F<:Union{MOI.VariableIndex,MOI.VectorOfVariables},S<:MOI.AbstractSet}
     return MOI.is_valid(model.optimizer, c)
 end
 
 function MOI.is_valid(
     model::Optimizer,
     c::MOI.ConstraintIndex{F,S},
-) where {F<:MOI.ScalarAffineFunction,S<:MOI.AbstractSet}
+) where {
+    F<:Union{
+        MOI.ScalarAffineFunction,
+        MOI.ScalarQuadraticFunction,
+        MOI.VectorAffineFunction,
+        MOI.VectorQuadraticFunction,
+    },
+    S<:MOI.AbstractSet,
+}
+    if haskey(model.constraint_outer_to_inner, c)
+        return true
+    end
     return MOI.is_valid(model.optimizer, c)
 end
 
diff --git a/test/jump_tests.jl b/test/jump_tests.jl
index 807ca42a..7a370a2a 100644
--- a/test/jump_tests.jl
+++ b/test/jump_tests.jl
@@ -427,11 +427,10 @@ function test_jump_set_variable_start_value()
     @variable(model, p in MOI.Parameter(0.0))
     set_start_value(x, 1.0)
     @test start_value(x) == 1
-    err = ErrorException(
-        "MathOptInterface.VariablePrimalStart() is not supported for parameters",
-    )
-    @test_throws err set_start_value(p, 1.0)
-    @test_throws err start_value(p)
+    @test_throws ErrorException(
+        "The parameter $(index(p)) value is 0.0, but trying to set VariablePrimalStart 1.0",
+    ) set_start_value(p, 1.0)
+    @test start_value(p) == 0.0
     return
 end
 
@@ -1275,14 +1274,51 @@ function test_parameter_Cannot_be_inf_2()
     return
 end
 
-function test_jump_psd_cone_with_parameter()
+function test_jump_psd_cone_with_parameter_pv()
     model = Model(SCS.Optimizer)
     @variable(model, x)
     @variable(model, p in MOI.Parameter(1.0))
-    @constraint(model, [[0 (p * x + -1)]; [(p * x + -1) 0]] in JuMP.PSDCone())
+    @constraint(
+        model,
+        con,
+        [[0 (p * x - 1)]; [(p * x - 1) 0]] in JuMP.PSDCone()
+    )
+    # the above constraint is equivalent to
+    # - (p * x -1)^2 >=0 -> (p * x -1)^2 <= 0 -> (p * x -1) == 0 -> p*x == 1
+    @test is_valid(model, con)
     optimize!(model)
     @test value(x) ≈ 1.0 atol = 1e-5
     set_parameter_value(p, 3.0)
     optimize!(model)
     @test value(x) ≈ 1 / 3 atol = 1e-5
 end
+
+function test_jump_psd_cone_with_parameter_pp()
+    model = Model(SCS.Optimizer)
+    @variable(model, x)
+    @variable(model, p in MOI.Parameter(1.0))
+    @constraint(
+        model,
+        con,
+        [[0 (x - p * p)]; [(x - p * p) 0]] in JuMP.PSDCone()
+    )
+    @test is_valid(model, con)
+    optimize!(model)
+    @test value(x) ≈ 1.0 atol = 1e-5
+    set_parameter_value(p, 3.0)
+    optimize!(model)
+    @test value(x) ≈ 9.0 atol = 1e-5
+end
+
+function test_jump_psd_cone_with_parameter_p()
+    model = Model(SCS.Optimizer)
+    @variable(model, x)
+    @variable(model, p in MOI.Parameter(1.0))
+    @constraint(model, con, [[0 (x - p)]; [(x - p) 0]] in JuMP.PSDCone())
+    @test is_valid(model, con)
+    optimize!(model)
+    @test value(x) ≈ 1.0 atol = 1e-5
+    set_parameter_value(p, 3.0)
+    optimize!(model)
+    @test value(x) ≈ 3.0 atol = 1e-5
+end