You need some legal document validly transferring the house from your parents to your. If they transfer the property to you by a quitclaim deed while they are alive that
is a valid legal transfer. But there may be problems with this method. First of all, legally they would no longer own their home. If you and they totally trust each other, that shouldn't be a problem.A possibly more difficult problem involves when they bought the house and how much it has increased in value (if it has).
If the house has increased significantly it value it can be desirable for you to inherit it after your parents have died. The reason for this is that inherited property gets what is called a "stepped-up basis" to its market value at death. I can't explain "stepped-up basis" in detail here, but let me give one example. Say the house was bought for $100,000, and its market value is now $300,00. If your parents quitclaim the house to you, you take it at their original basis, $100,000. If you sell the house for $300,000, you have a taxable profit of $200,000. But if you receive the house as an inheritance, after their deaths, the basis is "stepped-up" to $300,000. So if you sold it then, you would have no taxable profit.
On the other hand, there can be problems with waiting until death to inherit the house. If the property is lefty by a will, probate will be required. Probate is a lengthy and costly court process that will do you no good.
You say that you are "trustee". Trustee of what? Do your parents have a living trust? If they do, the house can be transferred by that trust after their death without any requirement of probate.
So, in sum, I suggest that you do some checking out of what your options are here, and which one seems best.