In a common-law situation if the parent of one of the parties involved purchases furniture for the couple who has the right to the said furniture?

First, common-law situations have gone out in many places. Many places no longer recognize common-law marriages. So ANY question on law needs to include your general locale as laws vary by area.

But, if the parents of a participant gave something to their child, its generally their child's property.

I will agree, where are you at, since at least in the US, there are few places that reconise it any longer.

Next if it does exist how was it created and what has been done to present it.

If they are spliting and it is a true common law, then they will have to do a divorce and property will be divided up in the divorce