Teaching experience as a subcontractor for a university program I work as an instructional staff at a local university. The program is marketed and hosted by the university and the certificates are granted by the university signed by the department dean as well. That being said, I was hired by and on the payroll of a subcontracting company, which also provides the curriculum.

I am also granted access to the university resources e.g. given a .edu email and listed under the university directory as an instructor-affiliate.

Does that mean I can list my experience as having worked for the university, the subcontracting company or both?
I suspect that legally you only work for the subcontractor since the university didn't hire you directly. This is especially true if the contractor supplies curriculum. 

But in the materials you can also say where you performed your duties, listing the university as the client, but not your employer. 

The university might object if they don't have a real relationship with you directly and you try to indicate otherwise.