diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index da9a87a6..ffd931a0 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -43,9 +43,16 @@ Fixes
 * **Mailjet:** Avoid a Mailjet API error when sending an inline image without a
   filename. (Anymail now substitutes ``"attachment"`` for the missing filename.)
   (Thanks to `@chickahoona`_ for reporting the issue.)
+
 * **Mailjet:** Fix a JSON parsing error on Mailjet 429 "too many requests" API
   responses. (Thanks to `@rodrigondec`_ for reporting the issue.)
 
+* **Postmark:** Fix a parsing error when Postmark indicates a sent message has
+  been delayed, which can occur if your message stream is paused or throttled or
+  when Postmark is experiencing service issues. These messages will now report
+  "queued" in the ``anymail_status`` (rather than throwing an error or reporting
+  "sent"). (Thanks to `@jmduke`_ for reporting the issue.)
+
 
 v12.0
 -----
diff --git a/anymail/backends/postmark.py b/anymail/backends/postmark.py
index 8c380622..6a445828 100644
--- a/anymail/backends/postmark.py
+++ b/anymail/backends/postmark.py
@@ -64,7 +64,8 @@ def parse_recipient_status(self, response, payload, message):
         for one_response in parsed_response:
             try:
                 # these fields should always be present
-                error_code = one_response["ErrorCode"]
+                # (but ErrorCode may be missing for Postmark service delays)
+                error_code = one_response.get("ErrorCode", 0)
                 msg = one_response["Message"]
             except (KeyError, TypeError) as err:
                 raise AnymailRequestsAPIError(
@@ -88,6 +89,11 @@ def parse_recipient_status(self, response, payload, message):
                         backend=self,
                     ) from err
 
+                status = "sent"
+                # "Message accepted, but delivery may be delayed." (See #392.)
+                if "delivery may be delayed" in msg:
+                    status = "queued"
+
                 # Assume all To recipients are "sent" unless proven otherwise below.
                 # (Must use "To" from API response to get correct individual MessageIDs
                 # in batch send.)
@@ -98,7 +104,7 @@ def parse_recipient_status(self, response, payload, message):
                 else:
                     for to in parse_address_list(to_header):
                         recipient_status[to.addr_spec] = AnymailRecipientStatus(
-                            message_id=message_id, status="sent"
+                            message_id=message_id, status=status
                         )
 
                 # Assume all Cc and Bcc recipients are "sent" unless proven otherwise
@@ -106,7 +112,7 @@ def parse_recipient_status(self, response, payload, message):
                 # original payload values.)
                 for recip in payload.cc_and_bcc_emails:
                     recipient_status[recip.addr_spec] = AnymailRecipientStatus(
-                        message_id=message_id, status="sent"
+                        message_id=message_id, status=status
                     )
 
                 # Change "sent" to "rejected" if Postmark reported an address as
diff --git a/tests/test_postmark_backend.py b/tests/test_postmark_backend.py
index 7d64f0f3..06f9730a 100644
--- a/tests/test_postmark_backend.py
+++ b/tests/test_postmark_backend.py
@@ -1079,6 +1079,22 @@ def test_mixed_response(self):
         self.assertEqual(status.recipients["valid@example.com"].status, "sent")
         self.assertEqual(status.recipients["spam@example.com"].status, "rejected")
 
+    def test_delivery_delayed_response(self):
+        # Postmark's "delivery may be delayed" response doesn't have an ErrorCode
+        # field set to 0 (despite their API docs).
+        response_data = {
+            "Message": "Message accepted, but delivery may be delayed.",
+            "MessageID": "38360f97-ff7f-44b2-bcd1-5ea94ff2af00",
+            "SubmittedAt": "2024-08-05T02:03:37.0951168Z",
+            "To": "to@example.com",
+        }
+        self.set_mock_response(json_data=response_data)
+        sent = self.message.send()
+        self.assertEqual(sent, 1)
+        status = self.message.anymail_status
+        self.assertEqual(status.recipients["to@example.com"].status, "queued")
+        self.assertEqual(status.message_id, "38360f97-ff7f-44b2-bcd1-5ea94ff2af00")
+
 
 @tag("postmark")
 class PostmarkBackendSessionSharingTestCase(