On the website of the Ministry of Foreign Affairs of Azerbaijan there are number of statements (like 1, 2, 3, 4, 5, etc; You'll find these statements if you go to the site and type
on removal of the name in the search box) reading:
On removal of the name of _____, citizen of _____, from the “List of
foreign citizens illegally visited to the occupied territories of the
Republic of Azerbaijan”.
In all of these statements, the ministry tells that it has considered the appeal from the traveler and decides to remove their name from the list (here is the latest version of the list), given that the traveler had a reasonable excuse. This means that it's even possible to visit the occupied territories and later to ask not to be included into this list, which is, however, strongly discouraged as it's a crime and is disrespectful to the territorial integrity of Azerbaijan and norms and principles of international law.
The first step for getting the permission would be to contact the secretariat of MFA:
- Address: Shikhali Gurbanov str. 50, Baku AZ 1009 Azerbaijan
- Phone: (+99412) 596 90 00
- Fax: (+99412) 596 90 01
- E-mail: firstname.lastname@example.org
You can also contact the Council of Appeal of MFA:
With all that being said, the ministry still doesn't advise to travel to occupied territories:
[...] the Ministry calls all foreign nationals to refrain themselves
from travelling to the occupied territories in and around the
Nagorno-Karabakh region of the Republic of Azerbaijan. [...]