There are a number of issues/questions here. 

First, have you already bought and closed on the property, or are you in the process of closing?  You can structure a Reverse 1031 Exchange if you have not already closed on the property.  However, if you have already closed on the property, then it gets more complicated. 

Let's take the related party issue first.  There was a comment that you can not buy/sell with a related party.  That is not entirely accurate. 

You can sell to a related party, but you will have a two (2) year holding period.  Your related party must hold the property they acquired from you for at least two (2) years before selling or they will trigger your capital gain taxes, and you will have to hold the replacement property that you acquire for at least two (2) years as well. 

Buying from a related party gets even more complicated, because you still have the two (2) year holding requirement, but in most cases it will not qualify because there is/will be basis swapping involved.  This issue is covered in IRS Revenue Ruling 2002-83, which you can read here: http://www.exeter1031.com/1031_exchange_revenue_ruling_2002_83.aspx.  This would most likely be the case based upon what you have said because you would be buying the property from your parents who would have no gain in the property (classic case of basis swapping).  This means that you would not be able to acquire from your parents.

Now let's take the ownership issue.  It is too late to structure any kind of 1031 Tax Deferred Exchange if you already own the property.  Transferring the property to your parents so that you can immediately buy it back from them as part of a 1031 Exchange would be considered a step transaction to avoid the payment of taxes and would not be permitted.  It could also be classified as tax evasion if the IRS could show that it was done on purpose with knowledge that it was wrong.