In SuiteCommerce Advanced, there is an extension that is working correctly on one domain, but it is throwing an error on another domain. The error is related to the service file. Both domains have separate SSP applications. What could be the issue?

Depends what the error is but my guess would be permissions on the service file.

The error is "Impossible to parse backend error", and i have checked the permission on the service file everything seems fine there.

That error is a generic error message when an SSP HTML page is sent back as the error. You will need to dig deeper into the error to see what the error is.

If its 2.0 check that you have a 2.0 SSP deployed to the domain getting the error

If you have scritable cart I would check that too