You don't have Time::Piece installed for /usr/bin/perl, so install it.
/usr/bin/perl -MCPAN -e install Time::Piece
You don't have DBI installed for /usr/local/bin/perl, so install it.
/usr/local/bin/perl -MCPAN -e install DBI
One more thing, in the Perl program, you have to manually tell it where to look. So I had to put this code line: use lib "/home/el/perl5/lib/perl5/x86_64-linux-thread-multi"; before the invocation of: use Time::Piece; in my perl program. I was able to find which path to use by running the command: locate Piece.pm
@Eric Leschinski, Your comment has nothing to do with this thread. You apparently installed a module into the non-default directory (and thus had to tell Perl where to look for it), but that's not what the OP is doing.
I understand now. The command you provided /usr/bin/perl -MCPAN -e install Time::Piece picks the wrong default installation directory and my perl couldn't find it. The user has to install it in the right place or else perl won't find it.