Why should I install both 32 and 64 bits Oracle provider for the same machine? I needed to read an Oracle DB from SQL Server, so I had to install the Oracle provider (ORAOLEDB.Oracle).

My machine:

System: Windows Server R2 2008 Enterprise, 64 bits (SP1).

RDMS: Microsoft SQL Server 2008 R2 (RTM) - 10.50.1600.1 (X64) 

However, at the beginning I installed just the 64 bits ODAC version for Oracle provider but it did not work. Finally I got it done by installing the 32 bits version, just how is said here.

Then I need to replicate the installation in production, I don't want to install both if is not required, so, I want to know why I should use both.
As alluded to in the question you linked to, some SQL Server tools/applications are 32-bit only. The main SQL Server application (sqlservr.exe) is 64-bit (32-bit may still be available as well, not certain about that for the latest versions); however, some of the supporting tools, perhaps most notable SQL Server Management Studio (ssms.exe), are only available as 32-bit applications.

If you want to install both versions, this should be sufficient reason.

If you want to find out if you can avoid installing the 64-bit version, all I can suggest is uninstalling that version from your development environment, and confirming whether or not everything still works as expected (including SSMS, and any other applications you use). If everything works fine, you can probably skip installing the 64-bit version in production. If you have problems, assume you need both versions.