diff --git a/airflow/kubernetes/pod_launcher_deprecated.py b/airflow/kubernetes/pod_launcher_deprecated.py
index 3367aa0cc23ef..ccf03b1e493c2 100644
--- a/airflow/kubernetes/pod_launcher_deprecated.py
+++ b/airflow/kubernetes/pod_launcher_deprecated.py
@@ -149,7 +149,8 @@ def monitor_pod(self, pod: V1Pod, get_logs: bool) -> Tuple[State, Optional[str]]
                 logs = self.read_pod_logs(pod, timestamps=True, since_seconds=read_logs_since_sec)
                 for line in logs:
                     timestamp, message = self.parse_log_line(line.decode('utf-8'))
-                    last_log_time = pendulum.parse(timestamp)
+                    if timestamp:
+                        last_log_time = pendulum.parse(timestamp)
                     self.log.info(message)
                 time.sleep(1)
 
@@ -174,7 +175,7 @@ def monitor_pod(self, pod: V1Pod, get_logs: bool) -> Tuple[State, Optional[str]]
             time.sleep(2)
         return self._task_status(self.read_pod(pod)), result
 
-    def parse_log_line(self, line: str) -> Tuple[str, str]:
+    def parse_log_line(self, line: str) -> Tuple[Optional[str], str]:
         """
         Parse K8s log line and returns the final state
 
@@ -184,7 +185,11 @@ def parse_log_line(self, line: str) -> Tuple[str, str]:
         """
         split_at = line.find(' ')
         if split_at == -1:
-            raise Exception(f'Log not in "{{timestamp}} {{log}}" format. Got: {line}')
+            self.log.error(
+                f"Error parsing timestamp (no timestamp in message: '{line}'). "
+                "Will continue execution but won't update timestamp"
+            )
+            return None, line
         timestamp = line[:split_at]
         message = line[split_at + 1 :].rstrip()
         return timestamp, message
diff --git a/airflow/providers/cncf/kubernetes/utils/pod_manager.py b/airflow/providers/cncf/kubernetes/utils/pod_manager.py
index 4221ac2afd95e..3b3a7b7a9eb9b 100644
--- a/airflow/providers/cncf/kubernetes/utils/pod_manager.py
+++ b/airflow/providers/cncf/kubernetes/utils/pod_manager.py
@@ -269,7 +269,11 @@ def parse_log_line(self, line: str) -> Tuple[Optional[DateTime], str]:
         """
         split_at = line.find(' ')
         if split_at == -1:
-            raise Exception(f'Log not in "{{timestamp}} {{log}}" format. Got: {line}')
+            self.log.error(
+                f"Error parsing timestamp (no timestamp in message '${line}'). "
+                "Will continue execution but won't update timestamp"
+            )
+            return None, line
         timestamp = line[:split_at]
         message = line[split_at + 1 :].rstrip()
         try:
diff --git a/tests/providers/cncf/kubernetes/utils/test_pod_manager.py b/tests/providers/cncf/kubernetes/utils/test_pod_manager.py
index e5526a73f94df..8070c3c3532b5 100644
--- a/tests/providers/cncf/kubernetes/utils/test_pod_manager.py
+++ b/tests/providers/cncf/kubernetes/utils/test_pod_manager.py
@@ -14,6 +14,7 @@
 # KIND, either express or implied.  See the License for the
 # specific language governing permissions and limitations
 # under the License.
+import logging
 from unittest import mock
 from unittest.mock import MagicMock
 
@@ -234,8 +235,11 @@ def test_parse_log_line(self):
         assert timestamp == pendulum.parse(real_timestamp)
         assert line == log_message
 
-        with pytest.raises(Exception):
+    def test_parse_invalid_log_line(self, caplog):
+        with caplog.at_level(logging.INFO):
             self.pod_manager.parse_log_line('2020-10-08T14:16:17.793417674ZInvalidmessage\n')
+        assert "Invalidmessage" in caplog.text
+        assert "no timestamp in message" in caplog.text
 
     @mock.patch("airflow.providers.cncf.kubernetes.utils.pod_manager.PodManager.run_pod_async")
     def test_start_pod_retries_on_409_error(self, mock_run_pod_async):