diff --git a/pkg/sql/logictest/testdata/logic_test/set_schema b/pkg/sql/logictest/testdata/logic_test/set_schema
index da2a1779a368..f2066be6ca6c 100644
--- a/pkg/sql/logictest/testdata/logic_test/set_schema
+++ b/pkg/sql/logictest/testdata/logic_test/set_schema
@@ -58,6 +58,16 @@ ALTER TABLE temp1 SET SCHEMA public
 statement ok
 INSERT INTO t VALUES (1)
 
+let $t_id
+SELECT 't'::regclass::oid::int
+
+# Verify the table's descriptor has the public schema as its parent.
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 'public' AND "parentID" = (SELECT id FROM system.namespace WHERE name = 'test' AND "parentID" = 0))
+  FROM crdb_internal.tables WHERE table_id = $t_id
+----
+true
+
 statement ok
 ALTER TABLE t SET SCHEMA s1
 
@@ -67,6 +77,13 @@ SELECT * FROM s1.t
 ----
 1
 
+# Verify the descriptor's parent_schema_id was updated to s1.
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 's1')
+  FROM crdb_internal.tables WHERE table_id = $t_id
+----
+true
+
 # Check that we can insert into the table.
 statement ok
 INSERT INTO s1.t VALUES (2)
@@ -90,6 +107,13 @@ SELECT * FROM public.t
 1
 2
 
+# Verify the descriptor's parent_schema_id was updated back to public.
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 'public' AND "parentID" = (SELECT id FROM system.namespace WHERE name = 'test' AND "parentID" = 0))
+  FROM crdb_internal.tables WHERE table_id = $t_id
+----
+true
+
 # Check that we can insert into the table.
 statement ok
 INSERT INTO public.t VALUES (2)
@@ -105,6 +129,9 @@ CREATE TABLE s1.t2(x INT)
 statement ok
 INSERT INTO s1.t2 VALUES (1)
 
+let $t2_id
+SELECT 's1.t2'::regclass::oid::int
+
 statement ok
 ALTER TABLE s1.t2 SET SCHEMA s2
 
@@ -113,6 +140,13 @@ SELECT * FROM s2.t2
 ----
 1
 
+# Verify the descriptor's parent_schema_id was updated from s1 to s2.
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 's2')
+  FROM crdb_internal.tables WHERE table_id = $t2_id
+----
+true
+
 # Ensure we cannot select from the old table.
 statement error pq: relation "s1.t2" does not exist
 SELECT * FROM s1.t2
@@ -520,3 +554,39 @@ ALTER TABLE "MixedCase"."DuplicateTable" SET SCHEMA "Another-Schema"
 user testuser
 
 subtest end
+
+# Verify that the table descriptor's parent_schema_id field is updated
+# correctly after `ALTER TABLE SET SCHEMA`.
+subtest regression_168255
+
+user root
+
+statement ok
+CREATE SCHEMA s_168255
+
+statement ok
+CREATE TABLE t_168255(x INT)
+
+statement ok
+ALTER TABLE t_168255 SET SCHEMA s_168255
+
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 's_168255')
+  FROM crdb_internal.tables
+ WHERE database_name = 'test' AND name = 't_168255'
+----
+true
+
+statement ok
+ALTER TABLE s_168255.t_168255 SET SCHEMA public
+
+query B
+SELECT parent_schema_id = (SELECT id FROM system.namespace WHERE name = 'public' AND "parentID" = (SELECT id FROM system.namespace WHERE name = 'test' AND "parentID" = 0))
+  FROM crdb_internal.tables
+ WHERE database_name = 'test' AND name = 't_168255'
+----
+true
+
+user testuser
+
+subtest end
diff --git a/pkg/sql/schemachanger/scexec/scmutationexec/create.go b/pkg/sql/schemachanger/scexec/scmutationexec/create.go
index 2ef4a02d003d..01d332664335 100644
--- a/pkg/sql/schemachanger/scexec/scmutationexec/create.go
+++ b/pkg/sql/schemachanger/scexec/scmutationexec/create.go
@@ -52,6 +52,9 @@ func (i *immediateVisitor) AddDescriptorName(ctx context.Context, op scop.AddDes
 		Name:           op.Namespace.Name,
 	}
 	i.AddName(op.Namespace.DescriptorID, nameDetails)
+
+	// TODO(bghal): This complementary operation to update the descriptor should
+	// be its own op.
 	if strings.HasPrefix(nameDetails.Name, catconstants.PgTempSchemaName) {
 		return nil
 	}
@@ -59,12 +62,11 @@ func (i *immediateVisitor) AddDescriptorName(ctx context.Context, op scop.AddDes
 	if err != nil {
 		return err
 	}
-
 	switch t := desc.(type) {
 	case *tabledesc.Mutable:
 		t.ParentID = op.Namespace.DatabaseID
-		t.UnexposedParentSchemaID = op.Namespace.SchemaID
 	}
+
 	return nil
 }
 
diff --git a/pkg/sql/schemachanger/scexec/scmutationexec/references.go b/pkg/sql/schemachanger/scexec/scmutationexec/references.go
index c0f5c2c560b5..9d05104758a6 100644
--- a/pkg/sql/schemachanger/scexec/scmutationexec/references.go
+++ b/pkg/sql/schemachanger/scexec/scmutationexec/references.go
@@ -11,6 +11,7 @@ import (
 	"github.com/cockroachdb/cockroach/pkg/sql/catalog"
 	"github.com/cockroachdb/cockroach/pkg/sql/catalog/descpb"
 	"github.com/cockroachdb/cockroach/pkg/sql/catalog/funcdesc"
+	"github.com/cockroachdb/cockroach/pkg/sql/catalog/tabledesc"
 	"github.com/cockroachdb/cockroach/pkg/sql/schemachanger/scop"
 	"github.com/cockroachdb/cockroach/pkg/sql/sem/catid"
 	"github.com/cockroachdb/cockroach/pkg/sql/sem/tree"
@@ -878,6 +879,10 @@ func (i *immediateVisitor) SetObjectParentID(ctx context.Context, op scop.SetObj
 			}
 		}
 		sc.AddFunction(obj.GetName(), ol)
+	case *tabledesc.Mutable:
+		t.UnexposedParentSchemaID = op.ObjParent.SchemaID
+	default:
+		return errors.AssertionFailedf("unexpected descriptor type %T for SetObjectParentID", obj)
 	}
 	return nil
 }