In AbstractMessageHandler.java is a function named createAttachmentsForMessage();

It's like this:

...
if (Part.ATTACHMENT.equalsIgnoreCase(disposition)) //there may be non-attachment parts (eg html email) Fixes JRA-1842

{ createAttachmentWithPart(part, issue); }

...

It handles only the ATTACMENT type attachments, but it shuld handle the INLINE attachments too.

Like this:
if (Part.ATTACHMENT.equalsIgnoreCase(disposition)||Part.INLINE.equalsIgnoreCase(disposition)) //there may be non-attachment parts (eg html email) Fixes JRA-1842

{ createAttachmentWithPart(part, issue); }