I fixed this issue after hearing back from Support. Basically, I just had to do a find/replace on the word "product" for the word "partner" (in this case, I based my module on the product module and my module is for "Partners"). Support specifically pointed me to the ParameterPermission Attribute that I decorated my DeletePartner method with, which I had defined as so:
, PartnersConstants.Security.PermissionSetName, PartnersConstants.Security.Delete)]
My "DeletePartner" method in my OpenAccessProvider then needed to have the attribute that is passed called "partner", whereas I had it defined as "product".
ORIGINAL DELETE PARTNER (Doesn't work):
NEWLY UPDATED DELETE PARTNER:
Once I did a find/replace in my project, it worked. Hopefully this information will be helpful to someone.