Should I run the new software system parallel with our old system?

While this approach was necessary in the days when systems were custom made for individual companies, current day software is much more reliable in its ability to process transactions.  Also given the high pace of most offices these days, running two systems for a significant length of time is just no longer practical for most companies.  Testing normally consists of running a test batch of transactions that represent the transactions that will be run on the system.  This is typically a couple of days to a weeks worth of transactions that give you the ability to evaluate the system and make any minor needed adjustments before going live.