Teradata ODBC - version 14 and 15 side by side
With regards to your question the readme specifies the following:


  TTU_BASE.15.10.12.00.readme.txt
  
  This release is not compatible with any Pre-15.10 release and it has
  to be removed  before installing this release.
  
  This release can co-exist with 16.00 and later releases.
  
  The TTU installer will block the installation if a pre-15.10 release
  is installed. Options:

  Continue (remove the pre-15.10 release first and then install the 15.10 release)

  Cancel the install

  
  Please see the Installation Guide for details.
  (https://teradatanet0.teradata.com/Site00BA)


But why keep 2 versions on the same server? TTU is 4 versions backwards compatible and 2 versions upwards compatible, so you are reasonable safe.

The biggest question is: is the software that is using the TTU certified for use of the TTU 15.10 version? I.e. Informatica Powercenter 9.6.1 is certified with TTU 14.10, but is NOT certified with TTU 15.10. In fact: I've found a knowledge base article from the vendor that specifically recommended with an error situation to first downgrade as they were not providing support on an not support combination of software.

In our case Powercenter 9.6.1 talks to TTU 14.10 and will be talking to Teradata 15.10. This combination is certified/supported.