There are a number of different issues here
"I'm now at a point of needing a care home for Mum, and probably Dad in the not too distant future. They have savings, and I'm aware the financial assessment will be done soon"
Presuming that your dad is over 60 and remains living in their home then, when the council do the financial assessment, they will disregard the value of the house when it comes to calculating your mum's capital.
So, unless, your mum has very large savings then it is unlikely that your mum will have to pay much or anything as long as your dad remains in their home.
The amount of capital taken into account varies between England, Wales and Scotland but, in England, it is £23,250 and higher elsewhere.
If your dad goes into a care home then the total value of the home will then be taken into account for both parents.
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"My parents owned a house, and for the last few years of living there put it in trust with me and my 2 siblings."
When it comes to protecting against care home fees this doesn't necessarily work. Depending on your circumstances and the attitude of the council they may decide that your parents had deliberately deprived themselves of an asset.
An important question about this is, were they then paying a market rent on the house to the trust or were they living in the house rent free?
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"Was the equity actually that of the 3 siblings, and so did we actually buy the new house, even though our names aren't on it?"
The equity belongs to the Trust and it is the responsibility of the trustees to manage the trust for the benefit of the named beneficiaries.
It is certainly possible for the terms of a trust to be written so that property can be sold and another property purchased using the trust funds.
If there was money left over from the sale of the old house and purchase of the new one then that money also belongs to the trust and should be managed for the benefit of the named beneficiaries - it does not belong to your parents.
However, because, of the way things have been done, the council may well decide that this was a deliberate "deprivation of capital" and count the money as still belonging to your parents.
I would really recommend that you speak to a solicitor experienced in this area. There are solicitors that are registered with the Society of Trust and Estate Practitioners (STEP) who specialise in these sorts of things.
If you google something like "STEP solicitor [name of your town]" then that should provide a list of local firms that have STEP registered solicitors.