Pl post OS details. Are you restoring the 10.2.0.3 backup to the 10.2.0.3 HOME ? Your error message indicates that you are trying to open the upgraded 11gR2 database using the 10.2.0.3 HOME.
ORA-00201: control file version 22.214.171.124.0 incompatible with ORACLE version 10.2.0.3.0
If you installed 11gR2 binaries in a separate folder/home, it should not overwrite the binaries in the 10.2.0.3 HOME.