diff --git a/pkg/internal/sqlsmith/BUILD.bazel b/pkg/internal/sqlsmith/BUILD.bazel
index 9eee5196ce69..46251e262c90 100644
--- a/pkg/internal/sqlsmith/BUILD.bazel
+++ b/pkg/internal/sqlsmith/BUILD.bazel
@@ -52,7 +52,6 @@ go_test(
         "//pkg/server",
         "//pkg/sql/parser",
         "//pkg/testutils/serverutils",
-        "//pkg/testutils/skip",
         "//pkg/testutils/sqlutils",
         "//pkg/testutils/testcluster",
         "//pkg/util/leaktest",
diff --git a/pkg/internal/sqlsmith/schema.go b/pkg/internal/sqlsmith/schema.go
index 46880a5c261e..1655545aa2ae 100644
--- a/pkg/internal/sqlsmith/schema.go
+++ b/pkg/internal/sqlsmith/schema.go
@@ -438,6 +438,14 @@ func (s *Smither) extractIndexes(
 		if err := rows.Err(); err != nil {
 			return nil, err
 		}
+		// Remove indexes with empty Columns. This is the case for rowid indexes
+		// where the only index column, rowid, is ignored in the SQL statement
+		// above, but the stored columns are not.
+		for name, idx := range indexes {
+			if len(idx.Columns) == 0 {
+				delete(indexes, name)
+			}
+		}
 		ret[*t.TableName] = indexes
 	}
 	return ret, nil
diff --git a/pkg/internal/sqlsmith/sqlsmith_test.go b/pkg/internal/sqlsmith/sqlsmith_test.go
index 59645b253f8e..a1ba39800249 100644
--- a/pkg/internal/sqlsmith/sqlsmith_test.go
+++ b/pkg/internal/sqlsmith/sqlsmith_test.go
@@ -21,7 +21,6 @@ import (
 	"github.com/cockroachdb/cockroach/pkg/ccl/utilccl"
 	"github.com/cockroachdb/cockroach/pkg/sql/parser"
 	"github.com/cockroachdb/cockroach/pkg/testutils/serverutils"
-	"github.com/cockroachdb/cockroach/pkg/testutils/skip"
 	"github.com/cockroachdb/cockroach/pkg/testutils/sqlutils"
 	"github.com/cockroachdb/cockroach/pkg/util/leaktest"
 	"github.com/cockroachdb/cockroach/pkg/util/randutil"
@@ -140,7 +139,6 @@ func TestRandTableInserts(t *testing.T) {
 // sometimes put them into bad states that the parser would never do.
 func TestGenerateParse(t *testing.T) {
 	defer leaktest.AfterTest(t)()
-	skip.WithIssue(t, 66723, "flaky test")
 	defer utilccl.TestingEnableEnterprise()()
 
 	ctx := context.Background()