No. The basic difference between a will and a Trust is that assets not in a trust that are to pass to your heir would have to be probated. The Trust avoids probate. When you receive your parents interest in their home, you will be able to claim a parent child exclusion from reassessment of property taxes.
As for Estate Taxes, because the exemption is now $5.25M per person, unless your parents estate is over that, or your estate is will be over that amount, then the issue between a Will and Trust are tax neutral.
Regardless, it will probably be worthwhile for you to sit down with an estate planning attorney either now or sometime in the future.
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These two situations appear to be totally unrelated. I do not see how YOUR executing a trust has anything to do with your inheritance from your parents' trust. There ARE advantages to having a trust, in many cases. Probate avoidance, privacy of the estate plan, flexibility of distributions and protection of minors are some of the advantages a trust provides compared to a Will. Tax avoidance is generally not something that a living trust can provide you.
I would consult with your own attorney to review this situation and determine the best planning tool for you to achieve your objectives.
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You may not have to pay taxes on the inheritance as the other attorney's have mentioned. Once you own "the property" the ordinary property taxes would apply. Your parents are right to urge to create a living trust so you can avoid probate fees.
The information above should not be considered legal advice and does not create an attorney-client relationship.
You don't identify what taxes that they are concerned that you might be liable for.... If your parents are concerned about estate taxes, it is their estate, not you, that is liable for payment of the taxes. If there is a concern about property taxes, in California there is a parent child exclusion from reassessment for property tax.
Perhaps they are urging you to have a trust because it will, if properly funded, avoid probate fees and costs at your death, just as they are attempting to avoid such fees and costs by having a trust (hopefully a funded one) in place when they eventually pass.
Almost assuredly, the cost of creating and funding a trust will be less than probate fees and costs.