Vitaly,
I have installed only the 32-bit SQL Native Client, but both the PeopleSoft 32-bit and 64-bit ODBC drivers. When Microsoft was assisting last night, they were able to confirm that the connection was using the 32-bit configuration on my Windows 2008R2 machine, and we confirmed that the settings are the same on the Windows Server 2012R2 machine.
The thing that you said here, though, that surprises me, was the "make sure it is SYSTEM DSN". On the Windows 2008R2 machine, the PeopleSoft PeopleTools driver by default is installed only under the USER DSN. When I try to connect to any database using the SQL Server Native Client, I can only see those registered in the USER DSN, I can't see any registered in the System DSN. Again, the installation on Windows 2008R2 works just fine. No SYSTEM DSN entries at all.