diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 89d8cc3..16a6e86 100644
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -2462,16 +2462,18 @@
         # Fix for SF #1100429: Under rare circumstances it can
         # happen that getmembers() is called during iteration,
         # which will cause TarIter to stop prematurely.
-        if not self.tarfile._loaded:
+
+        if self.index == 0 and self.tarfile.firstmember is not None:
+            tarinfo = self.tarfile.next()
+        elif self.index < len(self.tarfile.members):
+            tarinfo = self.tarfile.members[self.index]
+        elif not self.tarfile._loaded:
             tarinfo = self.tarfile.next()
             if not tarinfo:
                 self.tarfile._loaded = True
                 raise StopIteration
         else:
-            try:
-                tarinfo = self.tarfile.members[self.index]
-            except IndexError:
-                raise StopIteration
+            raise StopIteration
         self.index += 1
         return tarinfo
 
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index af59e27..69d342a 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -345,6 +345,14 @@
         finally:
             os.remove(empty)
 
+    def test_parallel_iteration(self):
+        # Issue #16601: Restarting iteration over tarfile continued
+        # from where it left off.
+        with tarfile.open(self.tarname) as tar:
+            for m1, m2 in zip(tar, tar):
+                self.assertEqual(m1.offset, m2.offset)
+                self.assertEqual(m1.name, m2.name)
+
 
 class StreamReadTest(CommonReadTest):