If you can create SQL Agent Jobs on the DB server, there is no logical reason why they should not let you load your SSIS package onto the DB server as it is after all just a big set of scripts at it's heart.
The DB server has to run the package, wherever it is stored and if you are forced to store it on the App server, then they will need to provide you with an account on the DB server that can access the app server over the network to read the package file.
If however, you explain to them that uploading the package into SSIS on the DB server is the alternative that means they do not need to provide and support the network account, they should say yes.
Either way, the job runs on the DB server, but the second way doesn't require network access privileges (unless the SSIS package requires them of course).
From the App Server, how can I execute SQL Server Agent jobs that reside on a remote DB Server
connect to SQL and EXEC sp_start_job passing it the job name.
which execute Packages that reside on the App Server?
Don't. Load them into SSIS on the DB server - they are just scripts, not applications...
But the larger question is how you totally separate SQL Database and SSIS on different servers when you have SQL 2012 on one server and only Visual Studio 2008 R2 on the other?
You don't. SSIS needs to be installed - you can't install it on your App server by your own comments....
MM
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