You are not experiencing an issue with TFAR (at least not as reported by your error message) but one with the CBA_A3 mod. As you are stating that you both got the mod off the workshop, then there are 2 things to check.
First is that you grabbed the right CBA_A3 file off the workshop, as for soem unknown reason, 14 other people have uploaded copies of the mod on the workshop under the same name as the original. The one you want is this one. The author that you'll see when searching for the mod on the workshop is "CBAteam" and that should be the same name that is reported in the Arma 3 launcher.
If you did download the correct mod, then it is possible that the mod got corrupted in downloading, and as such it no longer matches the server key that is checking it for authenticity. In this event, open your launcher and click on the "Mods" tab. Find the CBA_A3 mod in the list. Right click on the mod and look for the option to "Repair" the mod, which should be between the "Show dependencies" and "Remove mod" options. Click on Repair and see if that allows you in.
Best of luck!
