Looking at the log file, I found the following:
This is a OCM patch.
Home has OCM installed but not configured.
To run in silent mode, OPatch requires a response file for Oracle Configuration Manager (OCM).
Run /u01/app/oracle/product/oas10gr3/OPatch/ocm/bin/emocmrsp to generate an OCM response file. The generated response file can be reused on different platforms and in multiple OPatch silent installs.
To regenerate an OCM response file, rerun /u01/app/oracle/product/oas10gr3/OPatch/ocm/bin/emocmrsp.
ERROR: OPatch failed because of cmd. args. problem.
Doing a bit of digging, I found this recommendation to go ahead and run setupCCR in disconnected mode.
What's frustrating is that I requested that OCM not be installed or configured.
The following did the trick and I was able to complete the patch installation.
/u01/app/oracle/product/oas10gr3/ccr/bin/setupCCR -s -d