Reza Rad over at http://radacad.com/dateadd-vs-parallelperiod-vs-sameperiodlastyear-dax-time-intelligence-question explains the differences and provides easy usage scenarios for DateAdd, ParallelPeriod and SamePeriodLastYear DAX functions.

Read his blog post to see how he reached to below conclusion:

• DateAdd and SamePeriodLastYear both work based on the DYNAMIC period in the filter context.
• ParallelPeriod is working STATICALLY based on the interval selected in the parameter.
• ParallelPeriod and DateAdd can go more than one interval back and forward, while SamePeriodLastYear only goes one year back.
• DateAdd works on the interval of DAY, as well as month, quarter and year, but ParallelPeriod only works on month, quarter, and year.
• Depends on the filter context you may get a different result from these functions. If you get the same result in a year level context, it doesn’t mean that all these functions are the same! Look more into the detailed context.